Patent Publication Number: US-2019183196-A1

Title: Endless Neck Tie and Mobius Bow Tie

Description:
FIELD OF THE INVENTION 
     The present invention relates to apparel and, more specifically, to neck ties as well as methods of making and tying neck ties. Embodiments of the invention provide improved designs and methods of forming neck ties. 
     BACKGROUND OF THE INVENTION 
     Neckties and bowties provide flair and style for business, formal, and service dress wear. Improvements on the necktie and bowtie have focused on the construction of ties such as the bias tie disclosed in U.S. Pat. No. 1,447,090 of Langsdorf, issued 1923, the bow tie disclosed in U.S. Pat. No. 1,735,172 of Langsdorf, issued 1929, the necktie disclosed in U.S. Pat. No. 2,174,993 of Naftali, issued 1939, and the four-in-hand tie disclosed in U.S. Pat. No. 2,304,935 of Langsdorf, issued 1942; materials with which neckties are fabricated, such as to create the plastic bow tie disclosed in U.S. Pat. No. 2,578,269 of Sinton, issued 1951, and the leather bow tie disclosed in U.S. Pat. No. 1,692,991 of Howard, issued 1928; the symmetric bowtie disclosed in U.S. Pat. No. 1,737,072 of Blach, issued 1929; adjustable neckbands such as disclosed in U.S. Pat. Nos. 2,045,279 and 2,045,280 of Mayer, both issued 1936; decorative additions such as disclosed in U.S. Pat. No. 7,322,049 of Kennedy, issued 2006; and a multitude of pre-formed ties that allow the wearer to avoid learning how to tie a tie, including those disclosed in: U.S. Pat. No. 1,457,559 of Stonge, issued 1923, U.S. Pat. No. 1,540,037 of Smith, issued 1925, U.S. Pat. No. 2,061,630 of Keys, issued 1936, U.S. Pat. No. 3,034,134 of Peterson, issued 1962, U.S. Pat. No. 4,777,665 of Akamatsu, issued 1986, U.S. Pat. No. 4,856,115 of Knapp, issued 1989, U.S. Pat. No. 5,048,127 of Yang, issued 1990, U.S. Pat. No. 5,088,120 of Yen, issued 1991, U.S. Pat. No. 5,361,413 of Schaefer, issued 1991, U.S. Pat. No. 5,600,851 of McLeod, issued 1995, U.S. Pat. No. 5,6008,52 of Densch, issued 1997, and U.S. Pat. No. 6,691,319 of Simon, issued 2004. 
     Other than Langsdorf&#39;s innovations, these improvements have tended to focus on decorations and labor or time saving designs for neckties and bowties, perhaps at the expense of elegance. The scarf with a twist disclosed in U.S. Pat. No. 5,867,835 to Dabbieri (1999) which employs a “continuous circle of tubular fabric” (col. 2, line 21) with a 180-degree (col. 4, line 25) or 360-degree twist (col. 4, line 9) and specifies at col. 2, line 28, “A narrow neckband portion is then formed by accordion pleating and folding,” even emphasizes at col. 1, line 8, a scarf “that does not require any tying and/or closure manipulations whatsoever”. 
     Continued interest in new fashions and intriguing apparel creates a demand for innovation to provide flair and style in neck ties for business, formal, and service dress wear. Innovation in product design and appearance drives market trends, as evidenced in the fashion industry. There is always interest in new and intriguing apparel. Specifically, to generate higher demand for fashionable neck wear, it is beneficial to introduce new styles for neck ties in business, formal, and service dress. 
     SUMMARY OF THE INVENTION 
     In one series of embodiments an endless neck tie includes a strip of fabric of varying width formed as an endless loop having first and second sections. The width is measurable in one or more directions transverse to directions of endless travel along the strip of fabric. The first section has first and second end portions, and corresponds to a neckband when the endless loop is configured as a wearable neck tie. The first section has a narrow width at a position between opposing end portions thereof. The second section has first and second end portions. The first end portion of the second section extends toward the first end portion of the first section, and the second end portion of the second section extends toward the second end portion of the first section. The second section includes a blade portion having a wide width relative to the narrow width of the first section with which at least a first blade and a second blade of the wearable neck tie can be configured for presentation. The endless neck tie includes a first transition region and a second transition region along the first and second sections, the first transition region being about the first end portion of the first section and about the first end portion of the second section, and the second transition region being about the second end portion of the first section and about the second end portion of the second section. The first end portion of the first section and the first end portion of the second section have first variable positions definable along the first transition region by how the endless loop is configured about a neck as a neck tie. The second end portion of the first section and the second end portion of the second section have second variable positions definable along the second transition region by how the endless loop is configured about the neck as a neck tie. In one series of embodiments, when the endless loop is shaped as a circle, the position of the blade portion having the wide width along the strip of fabric is diametrically opposed to a position along the first section having the narrow width. The second section of the endless loop may include a first portion and a second portion each having a width wider than the narrow width of the first section so that, when the endless loop is configured as a wearable neck tie, the first blade is formed with the first portion of the second section and the second blade is formed with the second portion of the second section, and the first blade and the second blade each include a portion wider than a portion of the relatively narrow first section which forms the neck band. The strip of fabric formed as an endless loop may include a twist of 180 degrees which transforms the endless loop of fabric into a Mobius strip or may include a twist which is an integer multiple of 180 degrees. 
     According to another series of embodiments, an endless bow tie includes a strip of fabric of varying width formed as an endless loop for configuration as a wearable bow tie. The width is measurable in one or more directions transverse to directions of endless travel along the strip of fabric. The endless loop includes first and second variably definable sections. The first section has first and second end portions and corresponds to a neckband when the endless loop is configured as a wearable bow tie. The first section has a narrow width, Wn, at a position between opposing end portions thereof. The second section has first and second end portions, with the first end portion of the second section extending toward the first end portion of the first section, and the second end portion of the second section extending toward the second end portion of the first section. The second section includes a blade portion having a wide width, Ww, relative to the narrow width of the first section with which at least a first blade and a second blade of the wearable bow tie can be configured for presentation. The blade portion also includes a bridge region connecting the first blade and the second blade, the bridge having a width, Wb, narrower than the wide width of the blade portion. The bridge region demarcates the first blade from the second blade. 
     A bow tie shaped ornament is also provided. In one embodiment the ornament includes a generally elongate strip of material formed as an endless loop which is configured to define a neck band, at least first and second blades extending away from the neckband, and an outer knot positioned to demarcate a transition between the neck band and the blades. 
     A method is also provided for configuring an endless neck tie about a neck. According to one embodiment, a generally elongate length of fabric is provided in the form of an endless strip. A first segment and a second segment are defined along the endless strip by placing the endless strip around the neck and extending each segment along a different side of the neck and in front of the neck. The first segment and the second segment of the endless strip are brought adjacent one another or one over the other or otherwise in contact with one another in front of the neck. At least a first knot is formed proximal the front of the neck with a first sub-segment of the first segment and a first sub-segment of the second segment. A second sub-segment of each segment of the endless strip is positioned about the neck with each second sub-segment connected to the other second sub-segment near the nape of the neck. The two second sub-segments form a continuous neckband due to the continuous nature of the endless strip. A third sub-segment of each segment, corresponding to one or more blades, is extended away from the first knot, with each third sub-segment connected to the other third sub-segment. Portions of each segment are arranged to configure an endless neck tie such as an endless plumb neck tie or an endless bow tie. 
     According to another series of embodiments a method is provided for forming an endless tube by providing a first elongate strip of fabric and a second elongate strip of fabric, each strip having (i) a pair of opposing major edges extending along a first direction between first and second opposing strip ends, and (ii) a presentation side and a reverse side facing away from one another. The first and second strips of fabric are joined together by positioning the presentation side of each strip to face the presentation side of the other strip, with each major edge in one strip against a major edge in the other strip, to form the fabric into a tube shape. A pair of adjoining ends may be left unstitched to provide an open first tube end opposite a second tube end. According to one embodiment, the second tube end is extended through the open first tube end to turn the fabric tube to position the presentation sides of the first and second strips of fabric facing away from one another. The first and second opposing tube ends are joined to configure an endless tube. 
     A method is also provided for forming a bow tie with blades and without tabs, by providing a strip of material, and forming the strip into an endless loop. When the endless loop is tied into a bow tie, the bow tie has at least two blades and no tabs. In other embodiments, a bow tie comprises a strip of material in the form of an endless loop tied into a bow tie having at least two blades without any tabs. 
     A strap length adjustment system is also provided. In one embodiment, a first slide and a second slide each have a first channel and a second channel, each slide having first, second and third adjoining ribs, with the second rib positioned between the first rib and the third rib so that the first and second ribs define the first channel and the second and third ribs define the second channel. An end of a strap or an endless portion of a strap extends through the first channel of the first slide, then through the second channel of the first slide, then through the first channel of the second slide, then through the second channel of the second slide, then through the second channel of the first slide, then through the first channel of the first slide, then through the second channel of the second slide, and then through the first channel of the second slide. 
     A strap adjustment hardware slide according to an embodiment of the invention includes a first open channel and a second open channel, a first rib, a second rib, and a third rib. The first and second ribs define the first channel and the second and third ribs define the second channel. The first rib includes a first gaff spaced from the second rib to provide an opening into the first channel and the third rib includes a second gaff spaced from the second rib to provide an opening into the second channel. The second rib is connected with the first rib via a first slide neck, and the second rib is connected with the third rib via a second slide neck. 
     A method for adjusting a strap length includes providing a first slide and a second slide, each slide having first, second and third adjoining ribs, the second rib positioned between the first rib and the third rib so that the first and second ribs define a first channel and the second and third ribs define a second channel. Either an end of a strap or an endless portion of a strap is passed into the first channel of the first slide, then through the second channel of the first slide, then through the first channel of the second slide, then through the second channel of the second slide, then through the second channel of the first slide, then through the first channel of the first slide, then through the second channel of the second slide, and then through the first channel of the second slide. 
     According to another series of embodiments, a neck tie comprises a strip of material of varying width having first and second opposing ends connectable to one another to form an endless loop. The endless loop includes a neckband portion and a blade region portion. The neckband portion includes the connectable first and second opposing ends, having along part of its length a relatively narrow width suitable for placement about a neck. The blade region has a wide width relative to the narrow width of the neckband portion with which first and second blades are configurable. The blade region includes a juncture region along which the first and second blades are demarcated when the first and second blades are configured. The neckband portion includes an adjustment mechanism which variably overlaps part of the neckband portion with itself to adjust the length of the endless loop. 
     In yet another series of embodiments, a neck tie includes a strip of material of varying width having first and second opposing ends connected to one another to form an endless loop. The endless loop includes a neckband portion and a blade region. The neckband portion includes the connected first and second opposing ends, having along part of its length a relatively narrow width suitable for placement about a neck. The blade region has a wide width relative to the narrow width of the neckband portion with which first and second blades are configurable. The blade region includes a juncture region along which the first and second blades are demarcated when the first and second blades are configured. The strip includes a first flexible member extending toward the blade region and an adjustment mechanism comprising a second flexible member along part of the neckband portion, parts of which can be selectively included in the endless loop to adjust the length of the endless loop. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       These and other features of the present invention will become better understood when the following detailed description is read with reference to the accompanying drawings in which like characters represent like parts throughout, wherein: 
         FIGS. 1A and 1B  are, respectively, perspective front and rear views illustrating features of an endless plumb neck tie according to an embodiment of the invention; 
         FIGS. 2A and 2B  are perspective front views illustrating features of a Mobius plumb neck tie with, respectively, a twist at the outside knot and a twist on the front blade, according to embodiments of the invention; 
         FIG. 3A  is a perspective front view of an endless bow tie configured from an endless loop according to an embodiment of the invention; 
         FIG. 3B  is a view in cross-section of the endless bow tie shown in  FIG. 3A  taken along the endless loop; 
         FIGS. 3C, 3D, and 3E  are schematic knot diagrams for an endless bow tie from above, according to embodiments of the invention; 
         FIGS. 4A and 4B  are perspective front views illustrating features of embodiments of a Mobius bow tie, respectively, without and with contrasting fabrics; 
         FIG. 5A  is a perspective front view of an endless bow tie configured from an endless loop with double blades according to an embodiment of the invention; 
         FIG. 5B  is a view in cross-section of the endless bow tie shown in  FIG. 5A  taken along the endless loop; 
         FIG. 5C  is a perspective front view of the endless bow tie having double blades, as shown in  FIG. 5A , configured from an endless loop comprising contrasting fabrics; 
         FIGS. 6A through 6H  provide perspective views of Mobius bow tie ornaments according to embodiments of the invention; 
         FIGS. 7A through 7G  are front perspective views of a sequence of steps according to an overhand method to tie an endless plumb neck tie according to embodiments of the invention, with  FIG. 7H  further illustrating features of first and second sections of an endless loop shown in  FIG. 7B ; 
         FIGS. 8A through 8I  are front perspective views of a sequence of steps according to a simple loop method to tie an endless plumb neck tie according to embodiments of the invention; 
         FIGS. 9A through 9F  are front perspective views of a sequence of steps according to a four-in-hand method to tie an endless plumb neck tie according to embodiments of the invention; 
         FIGS. 10A through 10H  are front perspective views of a sequence of steps according to a simple loop method to tie a Mobius plumb neck tie according to embodiments of the invention, and  FIG. 10I  is a rear perspective view of the configured Mobius plumb neck tie; 
         FIGS. 11A through 11G  are views of a sequence of steps according to an overhand knot method for configuring an endless bow tie according to the invention, where  FIGS. 11A through 11C, 11F, and 11G  are front perspective views, and  FIGS. 11D and 11E  are schematic knot diagrams viewed from above; 
         FIGS. 12A through 12L  are views of a sequence of steps according to a square knot method for configuring an endless bow tie according to the invention, where  FIGS. 12A through 12I, 12K, and 12L  are front perspective views, and  FIG. 12J  is a schematic knot diagram viewed from above; 
         FIGS. 13A through 13L  are views of a sequence of steps according to a rotated knot method for configuring an endless bow tie according to the invention, where  FIGS. 13A through 13I, 13K, and 13L  are front perspective views, and  FIG. 13J  is a schematic knot diagram viewed from above; 
         FIGS. 14A through 14G  are views of a sequence of steps according to an overhand knot method for configuring a Mobius bow tie according to the invention, where  FIGS. 14A through 14C, 14F, and 14G  are front perspective views, and  FIGS. 14D and 14E  are schematic knot diagrams viewed from above; 
         FIGS. 15A through 15K  are views of a sequence of steps according to a square knot method for configuring a Mobius bow tie according to the invention, where  FIGS. 15A  through  15 I and  15 K are front perspective views, and  FIG. 15J  is a schematic knot diagram viewed from above; 
         FIGS. 16A through 16K  are views of a sequence of steps according to a rotated knot method for configuring a Mobius bow tie according to the invention, where  FIGS. 16A through 16I  and  FIG. 16K  are front perspective views, and  FIG. 16J  is a schematic knot diagram viewed from above; 
         FIGS. 17A through 17K  are views of a sequence of steps according to a square knot method for configuring an endless double bow tie according to the invention, where  FIGS. 17A through 17I  and  FIG. 17K  are front perspective views, and  FIG. 17J  is a schematic knot diagram viewed from above; 
         FIG. 18  is a perspective view of a first strip of fabric of varying width during an early stage of fabrication, patterned as a component for an endless plumb neck tie according to an embodiment of the invention; 
         FIGS. 19A through 19G  are perspective views illustrating the first strip of  FIG. 18  and another strip of fabric in an exemplary sequence of steps to form a tube which can be fashioned into an endless plumb neck tie; 
         FIGS. 20A through 20H  are views of an endless loop or Mobius loop of varying width according to embodiments of the invention, with comparable views of prior art, where: 
         FIGS. 20A and 20B  are, respectively, a top view in cross-section and an unrolled view of a prior art one-sided bow tie; 
         FIG. 20C  is a top schematic view in cross-section of an endless bow tie or Mobius bow tie according to an embodiment of the invention; 
         FIG. 20D  is an unrolled view of an endless untwisted loop or Mobius loop of varying width customized for a bow tie of  FIG. 20C ; 
         FIGS. 20E and 20F  are, respectively, a top view in cross-section and an unrolled view of a prior art ended bow tie; 
         FIG. 20G  is a top schematic view in cross-section of an endless untwisted double bow tie or Mobius double bow tie according to an embodiment of the invention; 
         FIG. 20H  is an unrolled view of an endless untwisted loop or Mobius loop of varying width customized for the double bow tie of  FIG. 20G ; 
         FIG. 21A  is a perspective view of a first strip of material of varying width during an early stage of fabrication, patterned as a component for an endless untwisted or Mobius bow tie according to an embodiment of the invention; 
         FIGS. 21B through 21G  are perspective views illustrating the first strip of  FIG. 21A  and another strip of material in an exemplary sequence of steps to form a tube which can be fashioned into an endless untwisted or Mobius bow tie; 
         FIGS. 22A, 22B, and 22C  are perspective front views of, respectively, a right-handed Mobius loop, a left-handed Mobius loop, and an untwisted endless loop, illustrating the chirality convention used to describe embodiments of the invention; 
         FIGS. 23A through 23I  are perspective front views of a partially fabricated endless neck tie according to an embodiment of the invention, illustrating a sequence of steps to join opposing tube ends of an endless untwisted loop or Mobius loop, with staggered widthwise ends; 
         FIGS. 24A and 24B  are front perspective views of an endless loop of varying width customized for, respectively, an endless untwisted plumb neck tie and a Mobius plumb neck tie; 
         FIGS. 25A and 25B  are front perspective views of an endless loop of varying width customized for, respectively, an endless untwisted bow tie and a Mobius bow tie; 
         FIG. 26  is a view of an endless plumb necktie incorporating an adjustment mechanism for the length of an endless neckband using slide hardware according to an embodiment of the invention; 
         FIGS. 27A through 27D  are views of a plumb necktie with integrally connected blades, each view incorporating an adjustment or closure mechanism according to an embodiment of the invention; 
         FIGS. 28A through 28D  are views of a bridged bow tie according to embodiments of the invention, each view incorporating an adjustment or closure mechanism; 
         FIGS. 29A and 29B  are perspective rear views of, respectively, an endless bow tie and an ended bow tie, each incorporating an adjustable-length neckband according to an embodiment of the invention; 
         FIGS. 30A through 30D  illustrate front views of length adjustment hardware suitable for use with neck bands of ended and endless neck ties according to embodiments of the invention, where: 
         FIGS. 30A and 30B  illustrate prior art hardware which may be used in a novel neckband adjustment arrangement shown in  FIG. 29 , and 
         FIGS. 30C and 30D  illustrate slide hardware according to embodiments of the invention, which may also be used in the novel neckband adjustment arrangement shown in  FIG. 29 ; and 
         FIGS. 31A-31C  illustrate an endless size adjustment system and method of providing length adjustments in a neckband of an endless neck tie or in a strap, where  FIG. 31A  is a perspective front view of the neckband or strap; and  FIGS. 31B and 31C  are views in cross-section of a neckband or strap taken along a horizontal plane, showing a method of providing length adjustment using slide hardware shown in  FIGS. 30A through 30D . 
     
    
    
     The drawings include standard drafting symbol patterns for representing color. The colors of the invention are not limited to those specifically indicated in the drawings. The exemplary color shading does not require color differentiation by. In accordance with common practice, the various described features are not necessarily drawn to scale, and may be drawn to emphasize specific features relevant to the invention. 
     DETAILED DESCRIPTION 
     The “Mobius Strip”, discovered by mathematician August Ferdinand Mobius (1790-1868), is a flat loop with a 180-degree twist. A Mobius strip (also known as a Mobius loop) can be made with a strip of material having two ends by twisting one end of the strip 180 degrees and attaching it to the other end of the strip. The Mobius strip is known for having a single surface and a single edge. 
     A study of various means of tying ended neckties and bowties led the inventor to study the class of tie knots using two successive overhand knots, such as the square knot of the bow tie. An abiding interest in the Mobius strip led the inventor to develop and experiment with a bowtie topology conforming to the Mobius strip. In embodiments comprising two contrasting fabrics, the inventor found that the two fabrics of differing color could be arranged to alternate colors from the first side of the neck band, to the first blade, to a first color of the knot, to a second color of the knot, to the second blade, and to the second side of the neck band. 
     Extending the study to plumb neckties led to an endless plumb necktie formed with a closed loop of fabric configured with one or more knots. Some embodiments of these endless plumb neckties feature knots only at the neckband, while others employ multiple knots affecting tie blade shape behind and near the bottom of the plumb necktie blade. Other shapes similar to those of a plumb necktie have been created by modifying traditional tie tying methods to take advantage of the closed nature of the endless or Mobius loop, for instance embodiments described herein of an endless plumb neck tie tied in a manner similar to the four-in-hand method of tying an ended plumb neck tie, or embodiments described herein of an endless bow tie tied in a manner similar to the square knot method of tying an ended bow tie. As used herein, the terms “neck tie” and “necktie” refer interchangeably to wearable neckwear including neck ties of traditional vertical configuration sometimes referred to as “four-in-hand” or “Langsdorf” or “standard” ties and herein called “plumb” neckties or “plumb” ties; and neck ties of bow tie horizontal configuration, unless otherwise specified or made clear from the context in which the term is used. The term “neckties and bowties” is used herein as an inclusive term (necktie) and a specific term (bowtie). Similarly, the term “endless” with regard to plumb neckties and bow ties refers to untwisted, Mobius, and other twisted embodiments of endless strips or loops, unless otherwise specified or made clear from the context in which the term is used. The term “ended” with regard to plumb neckties and bow ties refers to neck ties which are not endless. Prior art ended neck ties have ends which are referred to as tips, tabs, blade points, aprons, tails or, simply, ends. 
     “Twist” refers to a rotation transverse to the direction of endless travel along an endless loop, or transverse to the long dimension of a strip of fabric. Unless specified otherwise, “twist” refers to a “Mobius twist”, i.e. a 180-degree transverse rotation. 
     To facilitate understanding of the written description with respect to the drawings, although not limited to such, the term vertical can be regarded as the direction that extends downward or upward with respect to a ground plane, and the term horizontal refers to a direction that extends parallel with the ground plane. 
     Features of embodiments of my endless plumb necktie and endless bowtie include novel topologies and novel constructions relative to providing distinct and fashionable neck garb. Embodiments of the endless plumb necktie and endless bowtie provide added advantages of not slipping off the neck as conventional ended neckties and bowties sometimes do, and of having no ends to accidently slip through while tying. The endless and Mobius plumb neckties and bowties provide the advantage over pre-formed neckties and bowties of providing elegance, in fact, of providing mathematical elegance. 
     The mathematical concept of the Mobius strip is embodied in an endless plumb necktie or in an endless bow tie formed with a twisted endless loop made from a piece, a length, or a tube of material, referred to respectively as a Mobius plumb necktie or a Mobius bow tie. 
     An embodiment of an endless plumb neck tie comprises a strip of fabric joined at opposing ends to form an endless loop and folded to define a neck band comprising an elongate length of fabric, one or more blades, and at least one knot for encircling and constricting a region between the neck band and the blades. The term “strip” may refer to an elongate tube or length of fabric. 
     The term “neck” refers to any structure about which a neckband may be positioned, or, at least for purposes of illustration, a space about which such a structure may be placed, e.g., about which the neckband may be positioned. Exemplary structures include, but are not limited to, a human neck, a living creature neck, a mannequin neck, a collar, an animal collar, a wrist, a post, a pinback, a fastener, about which any neck band may be positioned, e.g., any of the neck bands  120 ,  220 ,  320 , or  420 . The exact nature of the neck or space is not limiting of the scope of the present invention. The term “nape” refers to the back of the neck. When the neck is shown as a space, the nape refers to a region of the space which would coincide with a nape if a neck were positioned in the space. The necktie is generally tied, the blades are pulled taut, and the necktie is arranged and straightened (“dressed”) at the front of the neck, a location on the neck opposite the region corresponding to the nape. 
     The term “neckband” refers to a ribbon-like element, i.e. a generally elongate strip, which is a segment or a section of an endless loop from which a necktie is formed. The neckband provides a means of positioning the necktie by encircling a relatively narrow region of a body, typically a neck. The exemplary embodiments illustrate neckbands as having relatively narrow widths compared to other portions of the endless loop, e.g., relatively wide blades of a plumb endless necktie. In many applications the neckband of a dressed tie is largely concealed by a collar and often need not be decorative since it is hidden from view. 
     Descriptions of the endless necktie and associated methods of manufacture or use are based on endless loops. Endless loops may be endless strips or endless tubes of material, e.g., fabric configurable as a wearable neck tie. A quantity of fabric, or other material, may be referred to as a length, as a piece, as a strip, or as a tube of fabric or material. A generally elongate strip of fabric has a relatively small width between two opposing “edges” along a direction in which a relatively large length extends, and also has two ends transverse to the direction in which the length extends. The terms “length of fabric” and “piece of fabric” both refer to a quantity of fabric having a relatively small width transverse to the direction in which a relatively large length extends. Referring to a quantity of fabric as a “piece of fabric” implies that the quantity of fabric may be a portion of a larger quantity of fabric, although that is not always the case, e.g., small hand-woven “pieces”. A piece of fabric has two opposing sides: normally a first side identified as the presentation side, i.e., the side intended to present upon completion of the necktie; and a second side identified as the reverse side, e.g., a side intended to face the interior of the fabric tube. Such a tube of fabric (or “fabric tube”) results from stitching two pieces of fabric together along two opposite edges while leaving two opposing ends not stitched. See, for example,  FIGS. 19 and 21  wherein two pieces of fabric are stitched with presentation sides together. An endless fabric tube results from stitching together the two opposing ends of a fabric tube, such as illustrated in  FIG. 23 . An endless fabric tube has a direction of endless travel which corresponds to the direction along which the length of fabric extends. 
     Portions of endless loops or tubes may be referred to as elements, regions, segments or sections. Segment and section demarcation may be arbitrary, e.g., resulting from how transitions between two adjoining portions in the endless loop (e.g., between a neck band and a blade portion) are defined as a tie is configured. Exemplary transition regions  179  in  FIG. 7H  may demarcate a transition between segments or sections at segment or section boundary regions, also referred to as “end regions”, and these transition regions are generally illustrated as transverse or perpendicular to a direction of endless travel along the loop. In  FIG. 7H , variable end regions  172  are shown for an exemplary blade section  170 , and variable end regions  178  are shown for an exemplary neckband section  724 . In one embodiment, the end regions  172 ,  178  are part of a transition region  179  variably positioned between a neckband section  724  and a section  170  corresponding to one pair of blades in a plumb neck tie. The example blade section  170  has four boundaries, i.e., the two end regions  172  which vary in location along the section  170 , and two opposing edge portions  174  which correspond to portions of the opposing edge portions in an endless loop of fabric and which terminate at the end regions  172 . A direction  176  of endless travel along the loop is shown in a direction which would traverse the end regions  172 ,  178 . Summarily,  FIG. 7H  illustrates a first section  724  of an endless plumb neck tie  100  corresponding to a neck band  120  having opposing end regions  178  which vary in location along the first section, and a second section  170  of an endless plumb neck tie  100  corresponding to a pair of blades  130 ,  140 , i.e., a blade portion having variable opposing end regions  172 . The endless neck tie includes a first transition region  179  and a second transition region  179  along the first and second sections, the first transition region  179  being about a first end portion  172  of the first section and about the first end portion  178  of the second section, and the second transition region  179  being about a second end portion  172  of the first section and about the second end portion  178  of the second section. Dashed lines are shown to indicate transitions  179  between end regions  172  and  178  of the adjoining neck band section and blade region section. It is to be understood that the locations of transitions  179  vary because the end regions  172  and  178  are variable for reasons now noted, e.g., depending, in part, on how much of the length of the endless loop is allotted to the blade pair section. From the foregoing description the term “transition” or “transition region” refers to an area where regions meet, and which may comprise, for example, portions of one or both end regions  172  and  178  of the regions which meet. 
     Those familiar with configuring ended neck ties will appreciate that neckbands and blade sections normally do not have fixed or predefined ends and, generally, do not have invariable lengths. Reference to a section or segment does not and should not connote portions having fixed or predefined ends. Neckbands and blade sections of endless neck ties have variable lengths so that someone configuring a neck tie has discretion to vary the length of the blade section and the neckband. Generally, the neckband length may vary depending on the neck size of the person wearing the apparel and the desired fit. For both plumb ties and bow ties, one blade section may be longer than the other by a discretionary amount to provide a desired appearance. The section or segment “end regions” thus are variable, while the terminology is necessary to describe divisions of the loop and the juxtaposition of various contiguous divisions of the loop. 
     The terms “blade” refers to each of one or more segments or sections in an endless loop from which a necktie is formed. For the embodiments corresponding to  FIGS. 1 and 2 , the blades of a dressed plumb necktie normally extend vertically downward from the front of the neck. For endless and Mobius bow tie embodiments such as illustrated in  FIGS. 3A, 4A, 5C, 11G, 12L, 13L, 14G, 15K, 16K, and 17K , the blades of a dressed tie normally extend horizontally along the front of the neck and away from an inside knot or an outside knot. Disclosed embodiments illustrate blades as wide expanses of fabric in an endless loop, relative to the width of an adjoining neckband. The term width refers to a neckband width, segment width, section width, or varying loop width. The terms wide and narrow are relative terms which refer generally to the dimension of the loop in the local plane of the loop transverse to a direction of endless travel along the loop. The blade is usually relatively wide, decorative, and nonfunctional other than stabilizing the knot or knots. The blade may be made from attractive, eye-catching, luxurious, or otherwise ornamental fabric. While the blades of a bow tie are conventionally referred to as “wings”, and the endless and Mobius bow tie blades may be referred to as “wings”, here the terms “blade” and “blades” are used with respect to the larger class of neckwear which includes the conventional ended plumb neck tie, bow tie and ascot, as well as the endless and Mobius plumb neckties and bow ties described herein. So, the set of all necktie blades includes the set of all bow tie wings. 
     The term “knot” refers to an element of a configured necktie which encircles and constricts a region of the necktie. In many embodiments the knot is made by crossing, tying or intertwining blades at a relatively narrow region of the necktie. The knot is usually both functional and decorative, and in many embodiments, the knot presents the same attractive, eye-catching, luxurious, or otherwise ornamental fabric as the blade or blades. An example of a knot is an outside knot that presents between a neck band and the tied blades. Formation of a knot often slightly folds or “dimples” the blade fabric adjacent the outside knot in plumb neckties and bow ties and embodiments of neckties formed with endless loops as illustrated in the figures. The dimple is the intentionally wrinkled region of a blade adjacent a knot resulting from constriction of the blade by the knot. Another example of a knot is an inside knot which may serve the purpose of securing the position of a neckband so an outside knot may be tied, resulting in the outside knot presenting, while the inside knot may be hidden behind the outside knot. 
     Prior art ended neckties of both the plumb and bow tie varieties may be made with a strip of material having two longer edges and two shorter ends. Ended bow tie blade ends are often referred to as “tabs”. The ends of an ended neck tie are often referred to as “aprons”. Thus the terms end, tab, and apron all may refer to an end or terminus region of a conventional ended plumb necktie or conventional ended bow tie made with a generally narrow strip. A feature which distinguishes endless or Mobius neck ties of this invention from a prior art ended plumb necktie or a prior art ended bow tie is the absence of ends in the sense that endless neckties do not have edges defining segment or section lengths but, instead, are continuous and unending loops. According to embodiments of the invention, one method of construction for an endless loop or Mobius loop of fabric from a tube of fabric, as illustrated in  FIGS. 18, 19, 20, 21, and 23 , provides staggered tube ends seamed to join opposing tube ends, resulting in an endless loop or Mobius loop The tube ends are intermediate, ephemeral features in the construction process and, while the seams persist in the finished necktie, the tube ends are not essential fabrication features of the fabric endless loop and are not functional components of the resulting tie. 
       FIG. 23  are perspective views of fabric strips stitched into tube shapes. The figures illustrate a sequence of steps by which opposing tube ends are joined to form an endless loop, with staggered widthwise ends, according to embodiments of the invention. 
     The term “bight” refers to a bend in a strip or loop of fabric formed without crossing one part of the strip or loop over another. The term “bight” is used herein to describe such a strip of fabric which has been bent or otherwise configured into a shape which defines a space partly bordered by the strip of fabric, the space referred to as a “bight opening” through which another part of the strip or loop of fabric may be passed. 
     The term “loop”, when referring to an endless loop or Mobius loop, is an embodiment of a shape that is circular or curved over on itself, respectively without or with a 180-degree twist. The terms “Mobius strip” and “Mobius loop” are interchangeable. Herein a loop is endless, whereas a strip may refer to a loop, e.g., a “Mobius strip” or a flat piece of fabric having two opposed ends. So, an endless loop may be made with a strip of material, having a first end and a second end, by attaching the first end of the strip to the second end of the strip. And, a Mobius loop may be made with a strip of material having a first end and a second end by twisting the first end of the strip 180 degrees and attaching it to the second end of the strip. 
     The term “loop” may be used as in common knot terminology, in which case a loop is a linear element such as a segment, section, blade, dual blade segment, or strip, which is folded or doubled upon itself so as to leave an opening. An “overhand loop” is created when a linear element is passed over an adjacent linear element, e.g., as described in the overhand knot method of tying illustrated in  FIG. 7  and particularly in  FIGS. 7E and 7F . The configured endless double bow tie  600  of  FIG. 5A  includes front loops  338 ,  348  and back loops  336 ,  346 . Thus the terms “simple loop method”, “overhand loop”, “underhand loop”, “blade loop”, “front loop”, and “back loop” clearly refer to common knot terminology, distinct from the terms “endless loop” and “Mobius loop”; the terms are distinguished by use of adjectives and by context. When used as a verb, “loop” refers to common knot terminology, as to fold or double a linear element upon itself so as to leave an opening. 
     The terms “tie” or “fold”, when used as verbs, refer to manipulating a strip or loop of fabric to create a pattern or shape by, for example, bending, overlapping, intertwining or knotting. The term “manipulate” means to handle, control, or maneuver. The term “tie”, when used as a verb, may refer to a sequence of steps which configure a strip or loop of fabric into a dressed necktie and, in particular, the term “tying” may used to describe forming or configuring an endless neck tie. 
     The term “staggered” refers to an arrangement of seams on adjacent fabrics such that the seams are not overlapped along their entire lengths, for purposes of minimizing the overall thickness or “bulk” of the necktie. Staggered seams inside of a neckband help reduce lumpy seams that might occur from adjacent fabrics stacked atop each other. 
     The terms “neck tie” and “necktie” are used interchangeably. The terms “bow tie” and “bowtie” are used interchangeably. The terms “Mobius strip” and “Mobius loop” are used interchangeably. The terms “neck band” and “neckband” are used interchangeably. The term “neck tie” refers both to plumb neck ties of traditional vertical configuration and to neck ties of bow tie horizontal configuration, unless otherwise specified or made clear from the context in which the term is used. 
       FIG. 1  illustrate an embodiment of an endless plumb neck tie  100  configured from an endless loop  110  (as shown elsewhere including  FIG. 7A ) as it would be worn about a neck  25  (not shown). A front view of the configured endless plumb tie is illustrated in  FIG. 1A  and a back/rear view of the configured endless plumb tie is shown in  FIG. 1B .  FIGS. 7, 8, and 9  each illustrate one of three different sequences for configuring the endless plumb neck tie  100  about a neck  25 , each sequence resulting in a different embodiment of an endless neck tie  100  configured with an endless loop  110 . 
     The configured endless plumb neck tie  100  of  FIG. 1  includes a neck band  120 , as it would be worn about a neck  25  (not shown), a first blade  130 , a second blade  140 , and an outside knot  160 . In some embodiments, the configured endless plumb neck tie  100  may also include an inside knot  150 . The outside knot  160  is shown in Fig lA where it constricts the juncture of the neck band  120  and first blade  130 . An inside knot  150  is shown in  FIG. 1B  where it constricts the juncture of the neck band  120 , first blade  130 , and second blade  140 .  FIGS. 1A and 1B  illustrate an embodiment which includes a juncture fold  184  corresponding to a transition between the first blade  130  and the second blade  140 . At a first position, P 1 , along the first blade  130 , a measurable first distance, D 1 , from the juncture fold  184 , the first blade  130  includes a first width, W 1 , which is wider than a second width, W 2 , of the second blade  140  at a second position, P 2 , along the second blade, a measurable second distance, D 2 , from the juncture fold  184 , where the first distance D 1  and the second distance D 2  are equal distances. The rate at which the first blade  130  tapers as a function of distance from the juncture fold  184  is less than the rate at which the second blade  140  tapers as a function of distance from the juncture fold  184 . 
     As shown in  FIGS. 7A, 8A, and 9A , each of three sequences of steps, by which an endless loop  110  is configured as it would be worn about a neck  25  (not shown) to form the endless plumb neck tie  100 , begins with placing an endless loop  110  about a neck  25  to define, with respect to the nape  35  of the neck  25 , a first segment  112  and a second segment  114  of the endless loop  110 . The segments  112 ,  114  each extend to an opposing side of the nape  35  and along different sides of the neck  25 . The first segment  112  and the second segment  114  of the endless loop extend from the nape  35  forward to contiguously meet one another at a juncture region  180  (shown in  FIGS. 7A, 8A, and 9A  positioned below the front  45  of the neck  25 ) due to the continuous nature of the endless loop. As shown in  FIGS. 7B, 8B, and 9B , the first segment  112  is then crossed over the second segment  114  to create a first crossover  116  to define the neckband  120 , comprising the first portion  112   a  of the first segment  112  from the front  45  of the neck to the nape  35  and the contiguous first portion  114   a  of the second segment  114  from the nape  35  to the front  45  of the neck. The first blade  130  includes the second portion  112   b  of the first segment  112  extending from the front  45  of the neck to the juncture region  180 . The second blade  140  includes the second portion  114   b  of the second segment  114  extending from the front  45  of the neck to the juncture region  180 . 
       FIG. 7  illustrate the “overhand” method of tying an endless plumb neck tie. As shown in  FIG. 7B , the first blade  130  is crossed over the second blade  140  at a crossover  116 . As shown in  FIG. 7C , a fold line  182  is made at the juncture region  180 , and a juncture fold  184  is made at fold line  182 . The fold line defines a transition between the first blade  130  and the second blade  140 . The fold line may be predefined in the endless loop  110  by, for example, thermal pressing or defining a line with a pattern of stitching. As shown in  FIG. 7D , the first blade  130  is brought near the second blade  140  along and near the juncture region  180 . Noting that the blades as shown in  FIG. 7D  are not drawn to scale,  FIG. 7E  illustrates the same blades  130 ,  140  lengthened to a scale suitable for illustrating several tying steps. The first blade  130  and the second blade  140  are brought into contact with one overlapping the other, followed by smoothing the blades together to create a dual blade segment  135 , which extends from the crossover  116  to the juncture region  180 . As also shown in  FIG. 7E , a bight  113  is formed by bending the dual blade segment  135  into a curved shape, thereby creating bight opening  115 . The juncture region  180 , as shown in  FIG. 7E , is then positioned above the bight  113  and the crossover  116 , in order to next pass the juncture region  180  through the bight opening  115 . As shown in  FIG. 7F , the juncture  180  is then passed or pulled through the bight opening  115  to fashion an outside knot  160  for the tie  100 , shown tightened in  FIG. 7G . That is, by pulling and extending the dual blade segment  135  vertically downward at the same time as controlling bight  113  to be proximal crossover  116 , the size of the bight opening  115  is reduced, and through tightening, bight  113  becomes outside knot  160 . Also, the action of pulling the dual blade segment  135  downward positions the bight  113  proximal the front  45  of the neck  25  so that with further pulling the outside knot  160  of the tie  100  is situated at the front  45  of the neck  25 . See, also, the illustrations of  FIG. 1 . The first blade  130 , the second blade  140 , and the outside knot  160  are then dressed. In this example, the resulting endless plumb neck tie  100  approximates the look of an ended plumb neck tie. 
       FIG. 8  illustrate the “simple loop” method of tying an endless plumb neck tie. As shown in  FIG. 8B , the first blade  130  is crossed over the second blade  140  at crossover  116 . As shown in  FIG. 8C , a fold line  182  is made at the juncture  180 , as described for the “overhand” method and shown in  FIG. 7C , and a fold  184  is made at fold line  182 . The fold line defines a transition between the first blade  130  and the second blade  140 . The fold line may be predefined in the endless loop  110  by, for example, thermal pressing or defining a line with a pattern of stitching. As shown in  FIG. 8D , the first blade  130  is brought near the second blade  140  along and near the juncture  180 . Noting that the blades as shown in  FIG. 8D  are not drawn to scale,  FIG. 8E  illustrates the same blades  130 ,  140  lengthened to a scale suitable for illustrating several tying steps. As shown in FIGS.  8 E,  8 F, and  8 G, the juncture  180  is pulled under the neckband  120  and then upward, inside then over the neckband  120 , and then above and past the crossover  116 . This pulling configures a portion of the second blade  140  into a bight  113 , and creates bight opening  115 . With continued pulling, the juncture region  180  is passed or pulled through the bight opening  115 , as shown in  FIGS. 8G and 8H , to fashion an outside knot  160  for the tie  100 , shown tightened in  FIG. 8I . That is, by pulling and extending juncture region  180  vertically downward at the same time as controlling bight  113  to be proximal crossover  116 , the size of the bight opening  115  is reduced, and through tightening, bight  113  becomes outside knot  160 . Also, the action of pulling the juncture region  180  downward positions the bight  113  proximal the front  45  of the neck  25  so that with further pulling the outside knot  160  of the tie  100  is situated at the front  45  of the neck  25 . See, also, the illustrations of  FIG. 1 . The first blade  130 , the second blade  140 , and the outside knot  160  are then dressed. In this example, the resulting endless plumb neck tie  100  of  FIG. 8I  approximates the look of an ended plumb neck tie. 
       FIG. 9  illustrate the “four-in-hand” method of tying an endless plumb neck tie. As shown in  FIG. 9B , the first blade  130  is crossed over the second blade  140  at a first crossover  116 . Noting that the blades as shown in  FIG. 9B  are not drawn to scale,  FIGS. 9C through 9F  illustrate the same blades  130 ,  140  lengthened to a scale suitable for illustrating several tying steps. As shown in  FIG. 9C , a fold line  182  is made at the juncture  180 , as described for the “overhand” method and shown in  FIG. 7C , and a fold  184  is made at fold line  182 . The fold line defines a transition between the first blade  130  and the second blade  140 . The fold line may be predefined in the endless loop  110  by, for example, thermal pressing or defining a line with a pattern of stitching. As also shown in  FIG. 9C , the first blade  130  is brought near the second blade  140  along and near the juncture  180 . The first blade  130  is next crossed under the second blade  140  at a second crossover  117 , and the first blade  130  is then crossed over the second blade  140  at a third crossover  118 . As shown in  FIG. 9D , the juncture region  180  is pulled under the neckband  120  and upward through the neckband  120 , while forming a portion of the second blade  140  into a bight  113 , thereby creating bight opening  115 . The juncture region  180  is then passed over the neckband  120  and then positioned over the bight opening  115  as shown in  FIG. 9E . With continued pulling, the juncture region  180  is passed or pulled through the bight opening  115 , to fashion an outside knot  160  for the tie  100 , shown tightened in  FIG. 9F . That is, by pulling and extending juncture region  180  vertically downward at the same time as controlling bight  113  to be proximal crossover  116 , the size of the bight opening  115  is reduced and, through tightening, bight  113  becomes outside knot  160 . Also, the action of pulling the juncture region  180  downward positions the bight  113  proximal the front  45  of the neck  25  so that with further pulling, the outside knot  160  of the endless plumb neck tie  100  is situated at the front  45  of the neck  25 . See, also, the illustrations of  FIG. 1 . The first blade  130 , the second blade  140 , and the outside knot  160  are then dressed. In this example, the resulting endless plumb neck tie  100  of  FIG. 9F  approximates the look of an ended plumb neck tie. The four-in-hand method of tying an endless plumb neck tie may be used to advantage with the endless loop  110  of  FIG. 9A  by further including a twist which is an integer multiple of 180 degrees (not shown), to compensate for twisting of the endless loop during the steps shown in  FIGS. 9B and 9C , as blades  130  and  140  are crossed over each other at crossovers  116 ,  117 , and  118 ; e.g., an endless loop with a left-hand twist of 540 degrees can provide a smooth appearance when the endless plumb neck tie is configured using the four-in-hand method of  FIG. 9 . 
       FIG. 2  illustrate an embodiment of a Mobius plumb neck tie  194  as it would be worn about a neck  25 . The Mobius plumb neck tie  194  is configured from a Mobius loop  190 , shown in  FIG. 10A , comprising a characteristic Mobius twist  192 .  FIGS. 2A and 2B  are both front views of the Mobius plumb neck tie  194 .  FIG. 2A  shows the Mobius twist  192  positioned in an outside knot  160  while  FIG. 2B  shows the Mobius twist  192  positioned in a first blade  130 . Multiple fabric color shadings and seam  124  are shown for the Mobius plumb neck tie  194  in  FIGS. 2A and 2B  to illustrate an exemplary alternation of fabrics. 
     The configured Mobius plumb neck tie  194  of  FIG. 2  includes a neck band  120 , a first blade  130 , a second blade  140 , an outside knot  160 , and a Mobius twist  192 . The configured Mobius plumb neck tie  194  may also include an inside knot  150  as shown in the rear perspective view of the Mobius plumb neck tie  194  of  FIG. 10I . The neck band  120  is shown in  FIG. 2  encircling a neck  25 . The outside knot  160  is shown in  FIGS. 2A  and  2 B positioned where it constricts the juncture of the neck band  120 , first blade  130 , and second blade  140 . 
     As shown in  FIG. 10 , a sequence of steps using the “simple loop” method by which a Mobius loop  190  is configured to form a Mobius plumb neck tie  194 , begins as in  FIG. 10A  with placing a Mobius loop  190  as it would be worn about a neck  25  (not shown) to define, with respect to the nape  35  of the neck  25 , a first segment  112  and a second segment  114  of the Mobius loop  190 . The segments  112 ,  114  each extend from an opposing side of the nape  35  and along different sides of the neck  25 . The segments  112 ,  114  meet contiguously at a juncture region  180  (shown positioned below the front  45  of the neck  25 ) due to the continuous nature of the Mobius loop. The first segment  112  and the second segment  114  of the endless loop extend from the nape  35  forward to meet one another near the front  45  of the neck  25  (shown in  FIG. 10A ). As shown in  FIG. 10B , the first segment  112  is then crossed over the second segment  114  to create a first crossover  116  to define the neckband  120 , comprising the first portion  112   a  of the first segment  112  from the front  45  of the neck to the nape  35  and the contiguous first portion  114   a  of the second segment  114  from the nape  35  to the front  45  of the neck. The first blade  130  includes the second portion  112   b  of the first segment  112  extending from the front  45  of the neck to the juncture region  180 . The second blade  140  includes the second portion  114   b  of the second segment  114  extending from the front  45  of the neck to the juncture region  180 . 
     With conformal positioning of the neckband  120  along the neck, the position of the Mobius twist  192  may be constrained to be located near the juncture region  180 , or on the first blade  130 , or as shown in  FIGS. 10B and 10C , on the second blade  140 . Crossover  116  is shown in  FIGS. 10B and 10C  where the first blade  130  is crossed over the second blade  140 . As shown in  FIGS. 10C and 10D , a fold line  182  is made at the juncture  180 , and a fold  184  is made at fold line  182 . As shown in  FIG. 10D , the first blade  130  is brought near the second blade  140  along and near the juncture  180 . Noting that the blades as shown in  FIG. 10D  are not drawn to scale,  FIG. 10E  illustrates the same blades  130 ,  140  lengthened to a scale suitable for illustrating several tying steps. As shown in  FIG. 10E , the juncture region  180  is pulled under the neckband  120  and then upward, inside then over the neckband  120 , and then above and past the crossover  116 . This pulling configures a portion of the second blade  140  into a bight  113 , thereby creating bight opening  115 . The juncture region  180  is passed over the neckband  120  as shown in  FIG. 10F . With continued pulling the juncture region  180  is passed or pulled through the bight opening  115 , as shown in  FIG. 10G , to fashion an outside knot  160  for the Mobius plumb neck tie  194  as shown in  FIG. 10H . That is, by pulling and extending juncture region  180  vertically downward at the same time as controlling bight  113  to be proximal crossover  116 , the size of the bight opening  115  is reduced and, through tightening, bight  113  becomes outside knot  160 . Also, the action of pulling the juncture region  180  downward positions the bight  113  proximal the front  45  of the neck  25  so that with further pulling, the outside knot  160  of the Mobius plumb neck tie  194  is situated at the front  45  of the neck  25 . See, also, the illustrations of  FIG. 1 . The first blade  130 , the second blade  140 , and the outside knot  160  are then dressed. In this example, the resulting Mobius plumb neck tie  194 , shown in the front perspective view of  FIG. 10H  and in the rear perspective view in  10 I, approximates the look of an ended plumb neck tie.  FIG. 10H  shows the Mobius twist  192  positioned in the outside knot  160 , although the Mobius twist  192  may be constrained to appear in any element of the neck tie, as noted in following paragraph describing some other embodiments of this invention. A feature of the Mobius plumb neck tie  194  made from two distinctive fabrics is the alternation of the fabrics in the dressed neck tie, as an exemplary arrangement, shown in  FIGS. 2A and 2B . 
     Embodiments of this invention include varying the simple loop method, overhand method, or the four-in-hand method of tying the Mobius plumb neck tie  194  by changing the configured location of the Mobius twist  192 . In one embodiment the Mobius twist  192  is constrained to the bight  113 , in which case the Mobius twist  192  may be positioned in the resulting outside knot  160  as shown in  FIGS. 2A and 10H . In another embodiment the Mobius twist  192  is constrained during tying to the region of the juncture region  180 , in which case the Mobius twist  192  may be unsecured and left to settle near the juncture region  180 ; or, the Mobius twist  192  may be moved to the first blade  130  as shown in  FIG. 2B , and further may be constrained to the middle of first blade  130  using standard tie hardware such as a tie pin or tie bar to better display contrasting fabrics. The Mobius twist  192  may be constrained to appear in any element of the neck tie, including the first blade  130 , the second blade  140 , the inside knot  150 , the outside knot  160 , and the neckband, for these alternate tying methods. Further, additional twists may be added in any element. For example, a 180-degree right-hand twist may be positioned in the neckband of an untwisted endless plumb necktie in order to create a corresponding 180-degree left-hand twist in the front blade to simulate the appearance of a Mobius plumb necktie. 
     An embodiment of an endless bow tie  200  according to the invention is illustrated in  FIG. 3 .  FIG. 3A  is a front view of the dressed bow tie and  3 B is a view in cross-section taken from above the bow tie  200 .  FIGS. 3C, 3D, and 3E  are schematic knot diagrams of the endless bow tie  200  viewed from above, illustrating, respectively, an overhand knot, a square knot, and a rotated knot. The rotated knot of  FIG. 3E  is formed with a variation of the square knot method. 
     The configured endless bow tie  200  of  FIGS. 3A and 3B  includes a neck band  220 , as it would be worn about a neck  25  (not shown), a first blade  230 , a second blade  240 , and an outside knot  260 . In some embodiments, the endless bow tie  200  may also be configured with an inside knot  250 . Endless loop  210 , shown in  FIGS. 11A, 12A, and 13A , is constricted by outside knot  260  as shown in  FIGS. 3A and 3B , about the juncture of the neck band  220 , the first blade  230 , and the second blade  240 . 
     As shown in  FIGS. 11A, 12A, and 13A , each of three sequences of steps, by which an endless loop  210  is configured as it would be worn about a neck  25  (not shown) to form one embodiment of the endless bow tie  200 , begins with placing an endless loop  210  about a neck  25  to define, with respect to the nape  35  of the neck  25 , a first segment  212  and a second segment  214  of the endless loop  210 . The segments  212 ,  214  each extend along an opposing side of the nape  35  and along different sides of the neck  25 . The segments  212 ,  214  meet contiguously at a juncture region  180  (shown positioned below the front  35  of the neck) due to the continuous nature of the endless loop. The first segment  212  and the second segment  214  of the endless loop extend from the nape  35  forward to meet one another near the front  45  of the neck  25  (shown in  FIGS. 11A, 12A , and  13 A). 
     Portions of the first segment  212  and the second segment  214  of the endless loop are conformed with the shape of the neck  25  as they extend from the nape  35  forward to (i) nearly meet near the front  45  of the neck  25  ( FIG. 11B ), or (ii) contact or cross over one another at a crossover  226  near the front  45  of the neck  25  ( FIGS. 12B and 13B ), to form the neckband  220 . As shown in  FIG. 11B , the first subsegment  212   a  of the first segment  212  extends from the front  45  of the neck to the nape  35 , and the first subsegment  214   a  of the second segment  214  extends from the nape  35  to the front  45  of the neck. As shown in  FIGS. 12B and 13B , the first subsegment  212   a  of the first segment  212  extends from the front  45  of the neck at a crossover  226  to the nape  35 , and the first subsegment  214   a  of the second segment  214  extends from the nape  35  to the front  45  of the neck, also at the crossover  226 . As shown in  FIGS. 11B, 12B, and 13B , the first blade  230  comprises a remaining portion, second subsegment  212   b , of the first segment  212  extending from near the front  45  of the neck at the crossover  226  to a juncture region  280 , and the second blade  240  comprises a remaining portion, second subsegment  214   b , of the second segment  214  extending from the front  45  of the neck at the crossover  226  to the juncture region  280 . The terms “bridge” or “bridge region” may be used to refer to juncture regions of relatively narrow width for which a first blade end region and a second blade end region terminate at opposed ends of the juncture region, such as juncture regions  280  and  380  for the endless bow tie. The terms “integrally connected blades” and “integrally connected blade region” may refer to a juncture region in an endless loop  210  that is of relatively wide width in, or about which, the first and second blades meet or are connected, such as juncture regions  180  and  780  for the endless neck tie. The terms “integrally formed blades” and “integrally connected blades” in an endless loop  210  refer to the presence of juncture regions at, or about which, the first and second blades meet or are connected. 
       FIG. 11  illustrate the “overhand knot” method of tying the endless bow tie. As shown in  FIG. 11C , a fold line  232  is made near the middle of the first blade  230 , and a fold  234  is made at fold line  232 ; a fold line  242  is made near the middle of the second blade  240 , and a fold  244  is made at fold line  242 . Noting that the blades as shown in  FIG. 11C  are not drawn to scale,  FIG. 11D  illustrates the same blades  230 ,  240  lengthened to a scale suitable for illustrating several tying steps. As shown in  FIG. 11D , the second blade  240  is crossed over the first blade  230  at a first crossover  216 . Noting that the blades as shown in  FIGS. 11A, 11B, 11C  are not drawn to scale,  FIGS. 11D and 11E  illustrate the same blades  230 ,  240  lengthened to a scale suitable for illustrating several tying steps. As shown in  FIG. 11E , the first fold  234  is crossed over the second blade  240  at a second crossover  217  and is then crossed under the second blade  240  at a third crossover  218 , forming an outside knot  260  near the front of the neck  45  ( FIG. 11F ). The first blade  230 , the second blade  240 , and the outside knot  260  are then dressed so the resulting endless bow tie  200  approximates the look of an ended bow tie ( FIG. 11G ). 
       FIG. 12  illustrate the “square knot” method of tying the endless bow tie and  FIG. 13  illustrate the “rotated knot” method of tying the endless bow tie. As shown in  FIGS. 12B, 12C, 13B and 13C , the second segment  214  is crossed over the first segment  212  at a first crossover  226 . 
     As shown in  FIGS. 12C and 13C , a fold line  232  is made near the middle of the first blade  230 , and a fold  234  is made at fold line  232 ; a fold line  242  is made near the middle of the second blade  240 , and a fold  244  is made at fold line  242 . As shown in  FIGS. 12D and 13D , the second blade  240  is pulled at or near the second fold  242  to cross the second blade  240  beneath the first crossover  226  and through the neckband  220 . This forms an inside knot  250 . Pulling of the second blade  240  continues until the juncture region  280  moves under the first crossover  226  ( FIGS. 12E, 12F, 13E, 13F ). As shown in  FIGS. 12G and 13G , the second fold  244  and second blade  240  are then crossed over the neckband  220 . As shown in  FIG. 12H , the second fold  244  and the second blade  240  are crossed over the first blade  230 , creating a second crossover  227 . Noting that the blades as shown in  FIG. 12H  are not drawn to scale,  FIG. 12I  illustrates the same blades  230 ,  240  lengthened to a scale suitable for illustrating several tying steps. As shown in  FIG. 12I , the first fold  232  is drawn over the second blade  240 , creating a third crossover  228 , and then passed under the second blade  240  to create a fourth crossover  229 , forming an outside knot  260 .  FIG. 12J  schematically illustrates the resulting outside knot  260 . As shown in  FIG. 12K , the first blade  230 , the second blade  240 , and the outside knot  260  are dressed so the resulting endless bow tie  200  approximates the look of an ended bow tie ( FIG. 12L ). 
       FIG. 13  illustrate the “rotated knot” method of tying the endless bow tie. As shown in  FIG. 13H , the second fold  242  and the second blade  240  are pulled under the first blade  230  at a second crossover  237 . Noting that the blades as shown in  FIG. 13H  are not drawn to scale,  FIG. 13I  illustrates the same blades  230 ,  240  lengthened to a scale suitable for illustrating several tying steps. As shown in  FIG. 13I , the second fold  242  is drawn over the first blade  230  at a third crossover  238 , then under the second blade  240  at a fourth crossover  239 , forming an outside knot  260 .  FIG. 13J  shows the corresponding knot diagram. As shown in  FIG. 13K , the first blade  230 , the second blade  240 , and the outside knot  260  are then dressed so the resulting endless bow tie  200  approximates the look of an ended bow tie ( FIG. 13L ). 
     A typical embodiment of a Mobius bow tie  300  of the present invention is illustrated in  FIG. 4 .  FIGS. 4A and 4B  are front views, both showing a Mobius twist  370  constrained to an outside knot  360 .  FIG. 4B  illustrates by color shading a possible alternation of fabrics and seam  324 . 
     The configured Mobius bow tie  300  of  FIG. 4  includes a neck band  320 , a first blade  330 , a second blade  340 , an outside knot  360 , and a Mobius twist  370 . The configured Mobius bow tie  300  may also include an inside knot  350 , as shown for example in  FIGS. 15J and 16J . In  FIG. 4 , the neck band  320  is shown as it would be worn about a neck  25  (not shown). The outside knot  360  is shown in  FIGS. 4A and 4B  constricting the juncture of the neck band  320 , first blade  330 , and second blade  340 . 
       FIGS. 14, 15, and 16  each illustrate a sequence of steps by which a Mobius loop  310  with Mobius twist  370  is configured as it would be worn about a neck  25  (not shown) to form an embodiment of a Mobius bow tie  300 . As shown in  FIGS. 14A, 15A , and  16 A, initially a Mobius loop  310  with Mobius twist  370  and seam  324  is placed over a nape  35  of a neck  25 , defining a first segment  312  and a second segment  314  of the Mobius loop  310 , which extend in opposite directions from the nape  35  and contiguously meet at a juncture region  380  due to the continuous nature of the Mobius loop  310 . T 
     The first segment  312  and the second segment  314  of the Mobius loop are conformed with the shape of the neck  25  from the nape  35  forward to the front  45  of the neck to define the neckband  320  (shown in  FIGS. 14B, 15B, and 16B ), comprising the portion  312   a  of the first segment  312  extending from the front  45  of the neck to the nape  35  and the contiguous portion  314   a  of the second segment  314  from the nape  35  to the front  45  of the neck. The first blade  330  comprises the portion  312   b  of the first segment  312  from the front  45  of the neck to the juncture region  380 . The second blade  340  comprises the portion  314   b  of the second segment  314  from the front  45  of the neck to the juncture region  380 . With the neckband  320  conformed with the shape of the neck  25 , the Mobius twist  370  may be constrained to be located at the juncture region  380 , or on the first blade  330 , or on the second blade  340 . A variation allows for the Mobius twist  370  to be constrained to be located on the neckband, in which case the Mobius bow tie may present the same as the endless bow tie of  FIGS. 3, 11, 12, and 13 . 
       FIG. 14  illustrate the “overhand knot” method of tying the Mobius bow tie. As shown in  FIG. 14C , a fold line  332  is made near the middle of the first blade  330 , and a fold  334  is made along fold line  332 ; a fold line  342  is made near the middle of the second blade  340 , and a fold  344  is made along fold line  342 . Noting that the blades as shown in  FIG. 14C  are not drawn to scale,  FIG. 14D  illustrates the same blades  330 ,  340  lengthened to a scale suitable for illustrating several tying steps. As shown in  FIG. 14D , the second blade  340  is crossed over the first blade  330  at a first crossover  316 . As shown in  FIG. 14E , the first blade  330  is crossed over the second blade  340  at a second crossover  317 , and the first blade  330  is crossed under the second blade  340  at a third crossover  318 , forming an outside knot  360  next to the front of the neck  45  ( FIG. 14F ). The first blade  330 , the second blade  340 , and the outside knot  360  are then arranged and straightened (“dressed”) so the resulting Mobius bow tie  300  approximates the look of an ended bow tie ( FIG. 14G ). 
       FIG. 15  illustrate the “square knot” method of tying the Mobius bow tie. As shown in  FIGS. 15B and 15C , the second blade  340  is crossed over the first blade  330  at a first crossover  326 . As shown in  FIG. 15C , a fold line  332  is made near the middle of the first blade  330 , and a fold  334  is made at fold line  332 ; a fold line  342  is made near the middle of the second blade  340 , and a fold  344  is made at fold line  342 . As shown in  FIG. 15D , the second fold  342  is pulled beneath and past the first crossover  326 , pulling the second blade  340  up through the neckband  320  and forming an inside knot  350 , until the juncture region  380  moves directly under the first crossover  326  ( FIGS. 15E, 15F ). As shown in  FIG. 15G , the second fold  344  and second blade  340  are drawn and then pulled down over the neckband  320 . As shown in  FIG. 15H , the second fold  342  and second blade  340  are pulled over the first blade  330  at a second crossover  327 . Noting that the blades as shown in  FIG. 15H  are not drawn to scale,  FIG. 15I  illustrates the same blades  330 ,  340  lengthened to a scale suitable for illustrating several tying steps. As shown in  FIG. 15I , the first fold  332  is drawn over the second blade  340  at a third crossover  328 , then under the first blade  330  at a fourth crossover  329 , forming an outside knot  360  next to the front of the neck  45 .  FIG. 15J  shows the corresponding knot diagram. As shown in  FIG. 15K , the first blade  330 , the second blade  340 , and the outside knot  360  are then arranged and straightened (“dressed”) so the resulting Mobius bow tie  300  approximates the look of an ended bow tie. 
       FIG. 16  illustrate the “rotated knot” method of tying the Mobius bow tie. As shown in  FIGS. 16B and 16C , the second blade  340  is crossed over the first blade  330  at a first crossover  356 . As shown in  FIG. 16C , a fold line  332  is made near the middle of the first blade  330 , and a fold  334  is made at fold line  332 ; a fold line  342  is made near the middle of the second blade  340 , and a fold  344  is made at fold line  342 . As shown in  FIG. 16D , the second fold  344  is pulled beneath and past the first crossover  356 , pulling the second blade  340  up through the neckband  320  and forming an inside knot  350 , until the juncture region  380  moves directly under the first crossover  356  ( FIGS. 16E, 16F ). As shown in  FIG. 16G , the second fold  342  and second blade  340  are drawn and then pulled down over the neckband  320 . As shown in  FIG. 16H , the second fold  342  and second blade  340  are pulled under the first blade  330  at a second crossover  357 . Noting that the blades as shown in  FIG. 16H  are not drawn to scale,  FIG. 16I  illustrates the same blades  330 ,  340  lengthened to a scale that illustrates several tying steps. As shown in  FIG. 16I , the second fold  342  is drawn over the first blade  330  at a third crossover  358 , then under the second blade  340  at a fourth crossover  359 , forming an outside knot  360 .  FIG. 16J  shows the corresponding knot diagram. As shown in  FIG. 16K , the first blade  330 , the second blade  340 , and the outside knot  360  are then dressed so the resulting endless bow tie  300  approximates the look of an ended bow tie. 
     A typical embodiment of an endless double bow tie  600  of the present invention is illustrated in the front view of  FIG. 5A  and in the top view of  FIG. 5B . A typical embodiment of the Mobius double bow tie  604  of the present invention is illustrated in the front view in  FIG. 5C . 
     The configured endless double bow tie  600  of  FIG. 5A  includes a neck band  320 , a front loop  338  of a first blade  330 , a back loop  336  of the first blade  330 , a front loop  348  of a second blade  340 , a back loop  346  of the second blade  340 , and an outside knot  360 . There may be an inside knot  350  (not shown) behind the outside knot  360 , depending on the tying method. The neck band  320  is shown encircling a neck  25 . The outside knot  360  is shown in  FIG. 5  where it constricts the juncture of the neck band  320 , first blade  330 , and second blade  340 . 
       FIG. 17  illustrate the “square knot” method of tying the endless double bow tie  600 .  FIG. 17  illustrate a sequence of steps by which an endless loop  602  is configured as it would be worn about a neck  25  to form an endless double bow tie  600 . As shown in  FIG. 17A , initially an endless loop  602  is placed over a nape  35  of a neck  25 , thereby defining a first segment  312  and a second segment  314  of the endless loop  602 , which segments extend from opposing sides of the nape  35  and contiguously meet at a juncture region  380  due to the continuous nature of the endless loop  602 . The first segment  312  and the second segment  314  of the endless loop are conformed to the shape of the neck  25  from the nape  35  forward to meet near the front  45  of the neck  25 . The neckband  320  (shown in  FIG. 17B ) comprises the portion of the first segment  312  extending from the front  45  of the neck to the nape  35  and the contiguous portion of the second segment  314  extending from the nape  35  to the front  45  of the neck. The first blade  330  comprises the portion of the first segment  312  from the front  45  of the neck to the juncture region  380 . The second blade  340  comprises the portion of the second segment  314  from the front  45  of the neck to the juncture region  380 . 
     As shown in  FIG. 17B , the first blade  330  is crossed over the second blade  340  at a crossover  326 . As shown in  FIG. 17C , a fold line  332  is identified near the middle of the first blade  330 , and a fold  334  is made at fold line  332 ; and a fold line  342  is identified near the middle of the second blade  340 , and a fold  344  is made at fold line  342 . As shown in  FIG. 17D , the second fold  342  is pulled beneath and past the crossover  326 , pulling the second blade  340  up inside the neckband  320  and forming an inside knot  350 , until the juncture region  380  moves directly under first crossover  326  ( FIGS. 17E and 17F ). As shown in  FIG. 17G , the second fold  342  and second blade  340  are drawn over the neckband  320 . As shown in  FIG. 17H , the second fold  342  is drawn over the first blade  330  at a second crossover  327 . Noting that the blades as shown in  FIG. 17H  are not drawn to scale,  FIG. 17I  illustrates the same blades  330 ,  340  lengthened to a scale suitable for illustrating several tying steps. As shown in  FIG. 17I , the first fold  332  is drawn over the second blade  340  at a third crossover  328  and under the second blade  340  at a fourth crossover  329 , forming an outside knot  360  next to the front of the neck  45 . As shown in  FIG. 17J , the first fold  332  is inverted by first unfolding it, then folding it in the other direction, and the inverted first fold  332  is then tucked between the inside knot  350  and the outside knot  360  to present as double blades, thereby defining a front loop  338  of the first blade  330  and a back loop  336  of the first blade  330 . Similarly, the second fold  342  is inverted by first unfolding it, then folding it in the other direction, and the inverted second fold  342  is then tucked between the inside knot  350  and the outside knot  360  to present as double blades, thereby defining a front loop  348  of the second blade  340  and a back loop  346  of the second blade  340 . 
     In one embodiment of the endless double bow tie  600 , notches  335  (shown in  FIG. 20H ) are located at fabric edges near fold line  332  between the back and front loops  336 ,  338  (shown in  FIGS. 17J, 20H ), and notches  345  (shown in  FIG. 20H ) are located at fabric edges near fold line  342  between the back and front loops  346 ,  348 . See  FIGS. 17J and 20H . Notches  335 ,  345  may help stabilize the positions of the inverted folds  332 ,  342  between the inside knot  350  and the outside knot  360  during wear. 
     The front loop  338  and back loop  336  of the first blade  330 , the front loop  348  and back loop  346  of the second blade  340 , and the outside knot  360  are then arranged and straightened (“dressed”) so the resulting endless double bow tie  600  approximates the look of an ended bow tie ( FIG. 17K ). The outside knot  360  constricts the juncture of the neck band  320 , the first blade  330  and the second blade  340 . 
     Front views of a Mobius double bow tie  604  are illustrated in  FIGS. 5A and 5C , both showing a Mobius twist  370  constrained to an outside knot  360 .  FIG. 5C  illustrates by color shading a possible alternation of fabrics and an associated seam  324 . The exemplary color shading does not require color differentiation. 
     The configured Mobius double bow tie  604  of  FIG. 5C  includes a neck band  320 , a front loop  338  of a first blade  330 , a back loop  336  of the first blade  330 , a front loop  348  of a second blade  340 , a back loop  346  of the second blade  340 , an inside knot  350 , an outside knot  360 , and a Mobius twist  370 . The outside knot  360  constricts the juncture of the neck band  320 , first blade  330 , and second blade  340 . 
     The Mobius double bow tie  604  is tied the same way as the endless double bow tie  600 , as shown in  FIG. 17 . Similar to the Mobius bow tie  300 , the Mobius twist  370  may be constrained to be located anywhere in the Mobius double bow tie  604 , including the neckband  320 , first and second blades  330  and  340 , inside and outside knots  350  and  360 , and juncture region  380 . There is great latitude in turning the blades right-side or inside out, in locating the Mobius twist, and in arranging the knots to present more or less fabric contrast. 
     Embodiments of a Mobius bow tie ornament  400  are illustrated in  FIG. 6 . The configured Mobius bow tie ornament  400  comprises a first blade  430 , a second blade  440 , an outside knot  460 , and a Mobius twist  470 . See  FIG. 6A .  FIG. 6B  shows an embodiment where the Mobius bow tie ornament is tied directly to a body, in this case a wrist. In some embodiments the Mobius bow tie ornament may be tied to a fastener. The term “fastener” refers to a means for attaching a bowtie or a bowtie ornament to a substrate such a clothing material, a body, hair, or an accessory. In the embodiment of  FIG. 6C , a pin fastener  411  attaches a decorative Mobius bow tie ornament  400  to clothing.  FIG. 6D  shows an embodiment in which a fastener includes a simple hooked closure  412 .  FIG. 6E  shows an embodiment in which a fastener is a carabiner clip  452 .  FIG. 6F  shows an embodiment in which a hair clip  414  called a “French clip” or “banana clip” acts as the fastener. In the embodiment of  FIG. 6G , the Mobius bow tie ornament  400  is fastened to a necklace  415 . In the embodiment of  FIG. 6H , the Mobius bow tie ornament is tied directly about a napkin  416 . Other embodiments include Mobius bow tie ornaments with or without fasteners arranged as hair bows, lapel pins, wrist bands, gift ribbon and wrap, and bracelets. 
     The Mobius bow tie ornament  400  is tied in a similar manner to the Mobius bow tie  300 , as shown in  FIG. 14  (the overhand knot method) and  FIG. 15  (the square knot method).  FIGS. 6C, 6D, 6E, 6F, and 6G  all show first and second blades parallel to the fastener structure  460 . This arrangement is best accomplished with the overhand knot method of  FIG. 14 .  FIGS. 6B and 6H  show first and second blades perpendicular to the body structure. This is best accomplished with the square knot method of  FIG. 15 . 
     An exemplary method for making an endless or Mobius loop with varying width customized for a plumb neck tie is illustrated in  FIGS. 18, 19, 22, and 23 . Embodiments of an endless neck tie may be fabricated from multiple strips of fabric formed into an endless tube. However, the invention is not so limited. For example, other embodiments may be fabricated with single strips of material, such as flat strips or extruded tubes that are flattened. Other embodiments comprise combinations of flat strips and tubes to form endless loops, and these may comprise different materials, including use of ribbon for the neckband and “patchwork” designs comprising discrete strips made from multiple pieces of fabric. 
       FIG. 18  illustrates a typical first strip of fabric  721  cut to be joined with a like second strip of fabric to make an endless or Mobius plumb neck tie. The first strip of fabric  721  includes a neckband section  724  and a blade region  726 . The neckband section  724  comprises a first neck band sub-section  731  and a second neck band sub-section  741 . Indicated portions of the blade region  726  correspond to a first blade  730  and a second blade  740 . The strip of fabric  721  includes first and second opposing ends  752 ,  754  which, when combined with corresponding ends of a like second strip  722 , as shown in  FIGS. 19B-19G , form first and second tube ends  762 ,  764 . See  FIGS. 19D-19G . A juncture region  780 , located in the blade region  726  demarcates portions corresponding to the first and second blades  730 ,  740 . 
       FIG. 19  illustrate a sequence of steps by which pieces of fabric are stitched and turned to form a fabric tube  710 , which may later be joined to make an endless or Mobius plumb neck tie by the method illustrated in  FIGS. 22 and 23 .  FIG. 19  further illustrate use of a turn assist  709 , shown here as a tied loop of thread, to reduce time spent coaxing the widest part of the turned tube to its fullest width for pressing. See  FIGS. 19D and 19E . 
     As shown in  FIG. 19A , initially a turn assist  709  is placed on the right side of a piece of fabric  721  at its widest part. As shown in  FIGS. 19B and 19C , a second piece of fabric  722  is placed right side down on the first piece of fabric  721 , sandwiching the turn assist  709  between the two pieces of fabric. As shown in  FIG. 19D , the two pieces of fabric are stitched together along elongate edges to form inside-out tube  710 , leaving the tube ends  752  and  754  free, and capturing the turn assist  709  in a first edge seam  714  and a second edge seam  716 . As shown in  FIG. 19E , the tube  710  is turned right-side out, with the turn assist  709  accessible on the outside of tube  710 . By pulling the turn assist  709 , the tube  710  is pulled wide for pressing, as shown in  FIG. 19F . With seams pressed and turn assist  709  cut away as shown in  FIG. 19G , tube  710  is complete and ready for joining of ends  752 ,  754  to make an endless or Mobius plumb neck tie by the method illustrated in  FIGS. 22 and 23 .  FIG. 19G  shows an exemplary embodiment of a tubular strip (prior to joining of the ends  752 ,  754 ) for an endless or Mobius loop with varying width, customized for endless and Mobius plumb neck ties, e.g. with variations in shape, width and length, as well as where and how to join tube ends  752 ,  754  to make a twisted or untwisted loop. 
     An exemplary method for making an endless or Mobius loop with varying width customized for endless and Mobius bow ties is illustrated in  FIGS. 20, 21, 22, and 23 .  FIGS. 20A and 20B  illustrate, respectively, a tied top view and an unrolled view of a prior art one-sided bow tie  390  which, when tied, has a tab  391  on one side of a knot and a blade  392  on the other side of the knot.  FIGS. 20C and 20D  are, respectively, a tied top view and an unrolled view of an endless or Mobius loop with varying width customized for endless and Mobius bow ties  200 .  FIGS. 20C and 20D  illustrate the endless bow tie  200  or Mobius bow tie  300  according to an embodiment of the invention which, when tied, has a first blade  330 , a second blade  340 , a juncture region  380 , an outside knot  360  and a neckband  320 . The neckband includes a first neckband portion  331 , a second neckband portion  341 , a first tube end  352  and a second tube end  354 .  FIGS. 20E and 20F  illustrate, respectively, a tied top view and an unrolled view of a prior art bow tie  395  which, when tied, has a tab  391  and a blade  392  on each side of a knot.  FIGS. 20G and 20H  illustrate, respectively, a tied top view and an unrolled view of the double endless bow tie  600  or double Mobius bow tie  604  which, when tied, presents as having pairs of blade loops ( 336 ,  338  and  346 ,  348 ) on each side of a knot  360 . 
     An ended prior art bow tie  395  comprising tabs  391  on each side, blades  392  on each side, a neckband  320 , and an outside knot  360 , is shown tied in the top view of  FIG. 20E  and in the unrolled view of  FIG. 20F . A double endless bow tie  600  and double Mobius bow tie  604 , according to embodiments of the invention are shown tied in the top view of  FIG. 20G  and in the unrolled view of  FIG. 20H . The bow tie  600  or  604  each, when tied, comprises a neckband, a first front blade  338 , a first back blade  336 , first notches  335 , a second front blade  348 , a second back blade  346 , second notches  345 , a juncture region  380 , and an outside knot  360 . The tied top view  FIG. 20G  illustrates the neckband  320 . With reference to the unrolled view of  FIG. 20H , the neckband includes a first neck band sub-section  331 , a first tube end  352 , a second neck band sub-section  341 , and a second tube end  354 . 
       FIGS. 21A-21G  illustrate a sequence of steps by which pieces of fabric are stitched and turned to form a fabric tube  810 , which may later be joined at opposing ends  352 ,  354  to make an endless or Mobius bow tie by the method illustrated in  FIGS. 22 and 23 .  FIG. 21  further illustrate the use of turn assists  709 , shown here as tied loops of thread, to reduce time spent coaxing the widest part of a turned fabric tube  810  to its fullest width for pressing, as shown in  FIGS. 21E and 21F . 
     As shown in  FIG. 21A , initially turn assists  709  are placed on the right side of a piece  821  of fabric at its widest parts, where piece  821  has been cut as in  FIG. 20D . As shown in  FIGS. 21B and 21C , a second piece of fabric  822  is placed right side down on the first piece of fabric  821 , sandwiching the turn assists  709  between the two pieces of fabric. As shown in  FIG. 21D , the two pieces of fabric are stitched along opposing elongate edges with presentation sides together to form inside-out tube  810 , leaving the tube ends  352  and  354  open, and capturing the turn assists  709  in a first edge seam  814  and a second edge seam  816 . A turn assist may be caught in one seam, e.g. blade  330  is shown with two short turn assists  709  each crossing one edge; or, a turn assist may be caught in more than one seam, e.g. blade  340  is shown with one long turn assist  709  crossing two edges. As shown in  FIG. 21E , the tube  810  is turned right-side out, with the turn assists  709  accessible on the outside of tube  810 . By pulling the turn assists  709 , the tube  810  is pulled wide for pressing, as shown in  FIG. 21F . With seams pressed and the turn assists  709  cut away as shown in  FIG. 21G , the tube  810  is complete and ready for joining of the ends  352 ,  354  to make an endless or Mobius bow tie by the method illustrated in  FIGS. 22 and 23 .  FIG. 21G  shows a typical embodiment of an endless or Mobius loop with variations in shape, width and length, as well as where and how to join ends  352 ,  354  of the tube to make a twisted or untwisted loop. 
       FIG. 22  illustrate the chirality (i.e., the “handedness”) of various endless and Mobius loops. In the United States a typical screw or bolt is likely to exhibit a “right-hand” orientation, that is, it has screw threads presenting as “\ \ \ \” (“backslash”). A right-hand screw is driven in a right-hand manner and is referred to as being right-handed. On the other hand, a screw or bolt from a toilet handle, from the left pedal of a bicycle, or from one half of a turnbuckle may exhibit a “left-hand” orientation, i.e., it has screw threads presenting as “/ / / I” (“forward slash”). A left-hand screw is referred to as being left-handed or having a “left-hand” orientation, and is driven in a left-handed manner. With this convention in mind, Mobius loops presenting as “==\==” are described as right-handed, and Mobius loops presenting as “==/ ==” are described as left-handed. Embodiments of the invention include both non-chiral (untwisted) loops and both right-handed and left-handed twists.  FIG. 22A  shows a right-handed Mobius loop  830  with a right-handed Mobius twist  832 ,  FIG. 22B  shows a left-handed Mobius loop  840  with a left-handed Mobius twist  842 , and  FIG. 22C  shows a non-chiral endless loop  850 . Individuals might favor left-handed or right-handed Mobius plumb neck ties or bow ties, and otherwise wonder why seemingly identical Mobius plumb neck ties or bow ties might differ in ease of use. 
       FIG. 23  illustrate a sequence of steps by which a fabric tube  510  may be configured to form an endless fabric tube or endless loop or Mobius loop  500 , selecting twist chirality from  FIG. 22 . While variation in width is not illustrated in fabric tube  510  in  FIG. 23 , the exemplary method shown in  FIG. 23  of making an endless fabric tube from a fabric tube applies as well to fabric tubes with varying width, such as fabric tube  710  of  FIG. 19G  and fabric tube  810  of  21 G. 
     Ends  552   a ,  552   b ,  554   a , and  554   b  are cut identically at a slant angle between  20  and 70 degrees, here illustrated as a slant angle θ of 45 degrees, with respect to the strip edge. By cutting all four ends identically, each end may accept stitching to an opposite end, regardless of whether right twist (as in  FIG. 22A ), left twist (as in  FIG. 22B ), or no twist (as in  FIG. 22C ) is applied to the strip.  FIG. 23A  shows an exemplary right twist  532  in fabric tube  510 .  FIG. 23A  shows ends  552   a  and  554   a  near each other, with their respective end seam lines  514   a  and  514   b  parallel to each other. Ends  552   b  and  554   b  are held out of the way.  FIG. 23A  shows the fabric reverse side of all four ends  552   a ,  552   b ,  554   a , and  554   b . End seam lines  514   a ,  514   b ,  516   a , and  516   b , are positioned on ends  552   a ,  554   a ,  552   a , and  554   a , respectively. Points where end seam lines intersect with edge seam lines provide effective “register marks” for positioning edges together for seaming; e.g., seam line intersection  512   a  is shown at the obtuse angle of end  552   a , seam line intersection  512   b  is shown at the acute angle of end  552   a , seam line intersection  512   c  is shown at the obtuse angle of end  554   a , and seam line intersection  512 d is shown at the acute angle of end  554   a . In the view of  FIG. 23B , end  554   a  is shown turned down so that the fabric presentation side of the end  554   a  is showing, with the presentation side of the end  554   a  facing the presentation side of the end  552   a . In this configuration, the edge of the end  554   a  is again positioned parallel to the edge of the end  552   a.    
     In  FIG. 23C , end  552   a  is translated over end  554   a , presentation sides facing each other, so that the seam line intersection  512   b  is positioned over intersection  512   c , and intersection  512   a  is positioned over intersection  512 d; and seam line  514   a  is positioned over seam line  514   b . Ends  552   a  and  554   a  are stitched together along seam lines  514   a  and  514   b , to make seam  514 .  FIG. 23D  shows seam  514  pressed open.  FIG. 23D  shows the fabric reverse side of all four ends  552   a ,  552   b ,  554   a , and  554   b . Seam line intersection  512   f  is shown at the obtuse angle of end  552   b , seam line intersection  512   e  is shown at the acute angle of end  552   b , seam line intersection  512   h  is shown at the obtuse angle of end  554   b , and seam line intersection  512   g  is shown at the acute angle of end  554   b.    
       FIG. 23E  shows ends  552   b  and  554   b  near each other, with their respective seam lines  516   a  and  516   b  parallel to each other. Ends  552   a  and  554   a  are pulled up and out of the way.  FIG. 23E  shows the fabric presentation side of ends  552   b  and  554   b . In the view of  FIG. 23F , end  554   b  is shown turned up so that the reverse side of the end  554   a  is showing, with the presentation side of the end  554   b  facing the presentation side of the end  554   b . In this configuration, the edge of the end  554   b  is again positioned parallel to the edge of the end  552   b . In  FIG. 23G , end  554   b  is translated over end  552   b , presentation sides facing each other, so that seam line intersection  512   g  is positioned over  512   f , and  512   h  is positioned over  512   e l ; and seam line  516     b  is positioned over  516   a . Ends  552   b  and  554   b  are stitched together along seam lines  516   a  and  516   b , to make seam  516 . Seam  516  is pressed open.  FIG. 23H  shows edge fabric turned to the inside of the loop along an edge seam line  518 , with the transformation of tube  510  to endless tube  500  nearly complete. Finishing stitches along edge seam line  518  complete loop  500 , as shown in  FIG. 23I . The method as illustrated results in seam  516  crossing seam  514  in a staggered or offset manner, to avoid an otherwise bulky seam. 
       FIGS. 24A and 24B  show, respectively, an endless loop  110  with varying width customized for an endless plumb neck tie  100  ( FIG. 24A ), and a Mobius loop  190  with varying width customized for a Mobius plumb neck tie  194  ( FIG. 24B ) with Mobius twist  192  and seam  124 .  FIGS. 25A and 25B  show respectively, an endless loop  210  with varying width customized for endless bow tie  200  ( FIG. 25A ), and a Mobius loop  310  with varying width customized for Mobius bow tie  300  ( FIG. 25B ) with Mobius twist  370  and seam  324 . When an endless loop such as loop  110 ,  190 ,  210 , or  310  is in the shape of a circle, as shown in the front perspective views in  FIGS. 24A, 24B, 25A, and 25B , a juncture position  128 , in the juncture region  180 ,  280 , or  380  along a blade portion  122  having a wide width along the endless loop, is diametrically opposed to a neckband position  126  along a neckband segment  120 ,  220 , or  320  having a narrow width relative to a blade portion  122 . The blade portion  122  includes the juncture region  180 ,  280 , or  380  (i) integrally formed with first and second blades  130 ,  140  of the second segment to meet with the blades  130 ,  140 ; or (ii) integrally formed with the first and second blades  230 ,  240  or  330 ,  340  of the second segment to terminate about the bridge  280  or  380  which includes a juncture position  128 . The width of the two blades may vary asymmetrically as a function of distance from the juncture position  128 . For numerous embodiments, the reference to blades as being integrally formed and the reference to a juncture region and blades as being integrally formed means that the components are formed along the same piece of material. 
     In  FIG. 25A , the neckband section  220  includes a narrow width, Wn, and the blade section  122  includes a wide width, Ww, relative to the narrow width Wn of the neckband section, with which at least a first blade and a second blade of the wearable bow tie can be configured for presentation, the blade section  122  also including the juncture region  280  also referred to as a bridge  280  connecting the first blade and the second blade. The bridge has a width, Wb, narrower than the wide width Ww of the blade portion, the bridge  280  demarcating the first blade from the second blade. 
       FIG. 26  is a view of an endless plumb necktie  100  comprising a combination of slide hardware with an endless neckband  120  to adjust the length of the endless neckband  120 , according to embodiments of the invention. Although this embodiment of the invention comprises two figure-8 slides  872  (referred to as slides  872   a  and  872   b ), other combinations of the slides shown in  FIG. 30  may be incorporated in this tie. The neck tie  100  includes a blade region  726  comprising material of a relatively wide width with which integrally connected first and second blades  730  and  740  are formed. The blade region  726  includes a juncture region  780  for variably demarcating the integrally connected first and second blades  730  and  740  from one another. The neckband  120 , also referred to as a neckband section  724 , includes a first neck band sub-section  731  which transitions to the first blade  730 , and a second neck band sub-section  741  which transitions to the second blade  740 . The portion of the neckband  120  corresponding to the second sub-section  741  passes through both openings in the first figure-8 slide  872   a , then passes through both openings in the second figure-8 slide  872   b , then once more passes through both openings in the first figure-8 slide  872   a , then once more passes through both openings in the second figure-8 slide  872   b , at which point the portion of the neckband corresponding to the first sub-section  731  transitions to the first blade  730 . 
     Although  FIGS. 27 through 29  illustrate plumb neckties comprising figure-8 slides  872  shown in  FIG. 30 , other slides shown in  FIGS. 30B-30D  may be used.  FIG. 27A  illustrates a plumb necktie  860  with integrally connected blades, with a T-hook (also referred to as T-hook hardware, or T-hardware) for neckband size adjustment. The neck tie  860  includes a blade region  726  comprising material of a relatively wide width with which integrally connected first and second blades  730  and  740  are formed. The blade region  726  includes a juncture region  780  for variably demarcating the integrally connected first and second blades  730  and  740  from one another. The neckband  120 , also referred to as a neckband section  724 , includes a first neck band sub-section  866  which transitions to the part of the blade region  726  corresponding to the first blade  730  of the configured neck tie  860 , and a second neck band sub-section  868  which transitions to the part of the blade region  726  corresponding to the second blade  740  of the configured neck tie  860 . 
     For purposes of illustrating the path of the neckband  120  passing through a ring  862  and into the T-hook  864 , the first neck band sub-section  866  is illustrated as a series of serially connected neck band sub-sections  866   a ,  866   b ,  866   c  and  866   d , and the second neck band sub-section  868  is illustrated as a series of serially connected neck band sub-sections  868   a  and  868   b . The first neck band sub-section  866   a  transitions from the first blade  730  to connect to a second neck band sub-section  866   b  which has size adjustment slits  863 . The second neck band sub-section  866   b  extends into the ring  862 , from which the third neck band sub-section  866   c  overlaps neck band sub-section  866   b  and extends into an opening in the T-hardware  864 , with the fourth neck band sub-section  866   d  looping back along the third neck band sub-section  866   c . The fourth neck band sub-section  866   d  may be stitched against or otherwise attached to the rear or front side of the third neck band sub-section  866   c  to form a closed loop. 
     In use, T-hardware  864  is engaged through a slit  863  chosen to customize the neck size of necktie  860 . Necktie  860  is an endless loop including the neckband section  865  with connectable opposing ends, first end  866  connected to second opposing end  868  via ring  862  at connection point  858 , including an adjustment mechanism which variably overlaps part of the neckband portion  866  with itself to adjust the length of the endless loop as the segment  866   c  overlaps neck band sub-section  866   b.    
       FIG. 27B  illustrates a plumb necktie  870  with integrally connected blades, with a figure-8 slide  872  for neckband size adjustment, and a blade region  726  corresponding to first and second blades  730  and  740  of the configured neck tie. The blade region  726  includes a juncture region  780  for variably demarcating integrally connected first and second blades  730  and  740 , from one another. The neckband  120 , also referred to as a neckband section  724 , includes a first neck band sub-section  876  which transitions to the part of the blade region  726  corresponding to the first blade  730  of the configured neck tie  870 , and a second neck band sub-section  878  which transitions to the part of the blade region  726  corresponding to the second blade  740  of the configured neck tie  870 . For purposes of illustrating the path of the neckband  120  through a ring  862  and the slide  872 , the first neck band sub-section  876  is illustrated as serially connected neck band sub-sections  876   a  and  876   b , and the second neck band sub-section  878  is illustrated as serially connected neck band sub-sections  878   a ,  878   b ,  878   c  and  878   d.    
     The first neck band sub-section  876   a  transitions from the first blade  730  to extend into the ring  862 , with the second neck band sub-section  876   b  looping back along the first neck band sub-section  876   a . The second neck band sub-section  876   b  may be stitched against or otherwise attached to the rear or front side of the first neck band sub-section  876   a . The second neck band sub-section  878  first neck band sub-section  878   a  transitions from the second blade  740  through first and second channels  940 ,  944  of the figure-8 slide  872 , from which the second neck band sub-section  878   b  extends through the ring  862 , from which the third neck band sub-section  878   c  overlaps neck band sub-section  878   b  and extends through one of the channels  940 ,  944  then through the other of the channels  940 ,  944  of the figure-8 slide  872 , with the fourth neck band sub-section  878   d  looping back along the third neck band sub-section  876   c . The fourth neck band sub-section  876   d  may be stitched against or otherwise attached to the rear or front side of third neck band sub-section  876   c  to form a closed loop. 
     In use, the figure-8 slide  872  is moved along neckband second portion  878  to customize the size of necktie  870 . Plumb necktie  870  is an endless loop including the neckband section  875  with connectable opposing ends, first end  876  connected to second opposing end  878  via ring  862  at connection point  858 , including an adjustment mechanism which variably overlaps part of the neckband portion  878  with itself to adjust the length of the endless loop as the segment  878   c  overlaps neck band sub-section  878   b.    
       FIG. 27C  illustrates a plumb necktie  880  with integrally connected blades, with a closure and chain for neckband size adjustment. The neck tie  880  includes a blade region  726  comprising material of a relatively wide width with which integrally connected first and second blades  730  and  740  are formed. The blade region  726  includes a juncture region  780  for variably demarcating the integrally connected first and second blades  730  and  740  from one another. The neckband  120 , also referred to as a neckband section  724 , includes a first neck band sub-section  886  which transitions to the part of the blade region  726  corresponding to the first blade  730  of the configured neck tie  880 , and a second neck band sub-section  888  which transitions to the part of the blade region  726  corresponding to the second blade  740  of the configured neck tie  880 . For purposes of illustrating the path of the neckband  120  through a ring  862 , also referred to as first ring  862   a  and second ring  862   b , the first neck band sub-section  886  is illustrated as a series of serially connected neck band sub-sections  886   a  and  886   b , and the second neck band sub-section  888  is illustrated as serially connected neck band sub-sections  888   a  and  888   b . The first neck band sub-section  886   a  transitions from the first blade  730  to extend into the first ring  862   a , with the second neck band sub-section  886   b  looping back along the first neck band sub-section  886   a . The second neck band sub-section  886   b  may be stitched against or otherwise attached to the rear or front side of first neck band sub-section  886   a  to form a closed loop. A flexible member such as a chain  882  is attached to ring  862   a.    
     The first neckband sub-section  888   a  of the second neckband sub-section  888  transitions from the second blade  740  to extend into the second ring  862   b , with the second neck band sub-section  888   b  looping back along the first neck band sub-section  888   a . The second neck band sub-section  888   b  may be stitched against or otherwise attached to the rear or front side of first neck band sub-section  888   a  to form a closed loop. A closure device, illustrated here as a lobster claw clasp  884 , is attached to ring  862   b . Ring  862   b , chain  882 , and lobster claw clasp  884  are part of neckband second portion  888 . In use, the lobster claw clasp  884  overlaps the chain  882  and is attached through a link of the chain  882  at a position so as to customize the neck size of necktie  880 . Plumb necktie  880  is an endless loop including the neckband section  885  with connectable opposing ends, first end  886  connected to second opposing end  888  via ring  862   a  at connection point  858 , including an adjustment mechanism which variably overlaps part of the neckband portion  878  with itself to adjust the length of the endless loop as the lobster claw clasp  884  overlaps chain  882 . 
       FIG. 27D  illustrates a plumb necktie  890  with integrally connected blades, with a hook-and-loop closure assembly comprising a hook portion  892  and a loop portion  894  for neckband size adjustment. The neck tie  890  includes a blade region  726  comprising material of a relatively wide width with which integrally connected first and second blades  730  and  740  are formed. The blade region  726  includes a juncture region  780  for variably demarcating integrally connected first and second blades  730 ,  740 , from one another. The neckband  120 , also referred to as a neckband section  724 , includes a first neck band sub-section  896  which transitions to the part of the blade region  726  corresponding to the first blade  730  of the configured neck tie  890 , and a second neck band sub-section  898  which transitions to the part of the blade region  726  corresponding to the second blade  740  of the configured neck tie  890 . The hook portion  892  is attached to neckband first sub-section  896  and the loop portion  894  is attached to neckband second sub-section  898 . The loop portion  894  is shown larger than the hook portion  892  so that, in use, the hook portion  892  may be variably positioned along the loop portion  894  so as to customize the size of necktie  890 . An additional loop portion may be attached on a second side (not shown) of necktie  890 , opposite loop portion  894 , allowing choice of chirality by providing the option to attach the hook portion  892  to either the first or second side of necktie  890 . Plumb necktie  890  is an endless loop including the neckband section  895  with connectable opposing ends, first end  896  connected to second opposing end  898  via hook-and-loop closure at connection point  858 , including an adjustment mechanism which variably overlaps part of the neckband  885  with itself to adjust the length of the endless loop as the hook-and-loop hook portion  892  overlaps hook-and-loop loop portion  894 . 
       FIG. 28A  illustrates a bridged bow tie  861  with a T-hook for neckband size adjustment. The bridged bow tie  861  includes serially-connected first blade  852  of a relatively wide width, bridge  850  of a relatively narrow width, and second blade  854  of a relatively wide width. The bridged bow tie  861  includes a neckband section  865  including a first neck band sub-section  866  which transitions to the first blade  852  of the configured bow tie  861 , and a second neck band sub-section  868  which transitions to the second blade  854  of the configured bow tie  861 . For purposes of illustrating the path of the neckband  865  through a ring  862  and into a T-hook  864 , the first neck band sub-section  866  is illustrated as a series of serially connected neck band sub-sections  866   a ,  866   b ,  866   c  and  866   d , and the second neck band sub-section  868  is illustrated as a series of serially connected neck band sub-sections  868   a  and  868   b . The first neck band sub-section  866   a  transitions from the first blade  852  to connect to a second neck band sub-section  866   b  which has size adjustment slits  863 . The second neck band sub-section  866   b  extends into the ring  862 , from which the third neck band sub-section  866   c  overlaps neck band sub-section  866   b  and extends into an opening in the T-hook  864 , with the fourth neck band sub-section  866   d  looping back along the third neck band sub-section  866   c . The fourth neck band sub-section  866   d  may be stitched against or otherwise attached to the rear or front side of the third neck band sub-section  866   c  to form a closed loop. In use, T-hook  864  is engaged through a slit  863  chosen to customize the neck size of necktie  860 . Although the T-hook  864  illustrated in  FIGS. 27A and 28A  may be referred to as hardware, composition of T-hooks used to practice the invention is not limited to any particular metallic or non-metallic material. The T-hook may, for example, be formed of a metal or a plastic. 
     Bow tie  861  is an endless loop including the neckband section  865  with connectable opposing ends, first end  866  connected to second opposing end  868  via ring  862  at connection point  858 , including an adjustment mechanism which variably overlaps part of the neckband portion  866  with itself to adjust the length of the endless loop as the segment  866   c  overlaps neck band sub-section  866   b.    
       FIG. 28B  illustrates a bridged bow tie  871  with a figure-8 slide for neckband size adjustment. The bridged bow tie  871  includes serially-connected a first blade  852  of a relatively wide width, a bridge  850  of a relatively narrow width, and a second blade  854  of a relatively wide width. The bridged bow tie  871  includes a neckband section  875  including a first neck band sub-section  876  which transitions to the first blade  852  of the configured bow tie  871 , and a second neck band sub-section  878  which transitions to the second blade  854  of the configured bow tie  871 . For purposes of illustrating the path of the neckband  865  through a ring  862  and the slide  872 , the first neck band sub-section  876  is illustrated as serially connected neck band sub-sections  876   a  and  876   b , and the second neck band sub-section  878  is illustrated as a series of serially connected neck band sub-sections  878   a ,  878   b ,  878   c  and  878   d . The first neck band sub-section  876   a  transitions from the first blade  852  to extend into the ring  862 , with the second neck band sub-section  876   b  looping back along the first neck band sub-section  876   a . The second neck band sub-section  876   b  may be stitched against or otherwise attached to the rear or front side of the first neck band sub-section  876   a . The first neck band sub-section  878   a  of second neck band sub-section  878  transitions from the second blade  854  through first and second channels  940 ,  944  of the figure-8 slide  872 , from which the second neck band sub-section  878   b  extends through the ring  862 , from which the third neck band sub-section  878   c  extends through one of the channels  940 ,  944  then through the other of the channels  940 ,  944  of the figure-8 slide  872 , with the fourth neck band sub-section  878   d  looping back along the third neck band sub-section  876   c . The fourth neck band sub-section  876   d  is stitched against or otherwise attached to the rear or front side of third neck band sub-section  876   c  to form a closed loop. In use, the figure-8 slide  872  is moved along neckband second portion  878  to customize the size of bridged bow tie  861 . Bow tie  871  is an endless loop including the neckband section  875  with connectable opposing ends, first end  876  connected to second opposing end  878  via ring  862  at connection point  858 , including an adjustment mechanism which variably overlaps part of the neckband portion  878  with itself to adjust the length of the endless loop as the segment  878   c  overlaps neck band sub-section  878   b.    
       FIG. 28C  illustrates a bridged bow tie  881  with a closure and chain for neckband size adjustment. The bridged bow tie  881  includes serially-connected a first blade  852  of a relatively wide width, a bridge  850  of a relatively narrow width, and a second blade  854  of a relatively wide width. The bridged bow tie  881  includes a neckband section  885  including a first neck band sub-section  886  which transitions to the first blade  852  of the configured bow tie  881 , and a second neck band sub-section  888  which transitions to the second blade  854  of the configured bow tie  881 . For purposes of illustrating the path of the neckband  885  through a first ring  862   a  and a second ring  862   b , the first neck band sub-section  886  is illustrated as serially connected neck band sub-sections  886   a  and  886   b , and the second neck band sub-section  888  is illustrated as serially connected neck band sub-sections  888   a  and  888   b . The first neck band sub-section  886   a  transitions from the first blade  852  to extend into the first ring  862   a , with the second neck band sub-section  886   b  looping back along the first neck band sub-section  886   a . The second neck band sub-section  886   b  is stitched against or otherwise attached to the rear or front side of first neck band sub-section  886   a  to form a closed loop. A flexible member such as a chain  882  is attached to the ring  862   a . The first neckband sub-section  888   a  of the second neckband sub-section  888  transitions from the second blade  854  to extend into the second ring  862   b , with the second neck band sub-section  888   b  looping back along the first neck band sub-section  888   a . The second neck band sub-section  888   b  is stitched against or otherwise attached to the rear or front side of first neck band sub-section  888   a  to form a closed loop. A closure device, illustrated here as a lobster claw clasp  884 , is attached to the ring  862   b . The ring  862   b , the chain  882  and the lobster claw clasp  884  are part of the neckband second portion  888 . In use, the lobster claw clasp  884  captures a link in the chain  882 , being attached through a link of the chain  882  selected at a chain position so as to customize the neck size of bridged bow tie  881 . Bow tie  881  is an endless loop including the neckband section  885  with connectable opposing ends, first end  886  connected to second opposing end  888  via ring  862   a  at connection point  858 , including an adjustment mechanism which variably selects part of the chain  882  with the lobster claw clasp  884  to adjust the length of the endless loop. 
       FIG. 28D  illustrates a bridged bow tie  891  with a hook-and-loop closure assembly, comprising hook portion  892  and a loop portion  894  for neckband size adjustment. The bridged bow tie  891  includes serially-connected a first blade  852  of a relatively wide width, a bridge  850  of a relatively narrow width, and a second blade  854  of a relatively wide width. The bridged bow tie  891  includes a neckband section  895  including a first neck band portion  896  which transitions to the first blade  852  of the configured bow tie  891 , and a second neck band portion  898  which transitions to the second blade  854  of the configured bow tie  891 . The first blade  852  is attached to the neckband first portion  896  of the configured bow tie  891 , and the second blade  854  is attached to the neckband second portion  898  of the configured bow tie  891 . The hook-and-loop closure assembly is attached to the neckband first and second portions  896  and  898 , such that one neckband portion receives the hook portion  892  of hook-and-loop closure assembly and the other neckband portion receives the loop portion  894  of hook-and-loop closure assembly.  FIG. 28D  shows the hook portion  892  attached to neckband first portion  896  on the first side of bow tie  891 , and the loop portion  894  attached to neckband second portion  898  on the first side of bow tie  891 . The loop portion  894  is shown larger than the hook portion  892  so that, in use, the hook portion  892  may be adjustably positioned along the loop portion  894  at a connection point  892  to customize the size of the bow tie  891 . An additional loop portion may be attached on the second side of bow tie  891 , opposite the loop portion  894 , allowing choice of chirality by providing the option to attach the hook portion  892  to the loop portion on either the first or second side of necktie  891 . The bow tie  881  is an endless loop including the neckband section  895  with connectable opposing ends, first end  896  connected to second opposing end  898  via the hook-and-loop closure assembly at the selectable connection point  858 . This adjustment mechanism variably overlaps part of the neckband  885  with itself to adjust the length of the endless loop, securing the selected length by attaching the hook portion  892  to the hook-and-loop loop portion  894 .  FIGS. 29A and 29B  are perspective rear views of, respectively, an endless bow tie  300  and an ended bow tie  920 , each with an endless length adjustment system  900  comprising an arrangement of exemplary hardware slides  872  to adjust neckband length. See, also,  FIG. 26 . The embodiments of  FIG. 29  illustrate two figure-8 slides  872   a  and  872   b , but any combination of the slide hardware components  872 ,  926 ,  928  and  930 , shown in  FIG. 30 , may be employed in the adjustment system  900 . The endless bow tie  300  of  FIG. 29A  includes a first blade  330 , an inside knot  350  or an outside knot  360 , a second blade  340 , a first reference point  337  on the neckband  320 , positioned near the blade  330 , and a second reference point  347  on the neckband  320 , positioned near the blade  340 . The ended bow tie  920  of  FIG. 29B  includes a first tab  391   a , a first blade  392   a , a first reference point  337  on the neckband  320 , positioned near the blade  392   a , a second tab  391 b, a second blade  392   b , and a second reference point  347  on the neckband  320 , positioned near the blade  392   b . For both of the bowties  300  and  920 , for purposes of illustrating the path of the neckband  320  through the two figure-8 slides  872   a  and  872   b , the neck band  320  is illustrated as a series of serially connected neck band sub-sections  320   a ,  320   b ,  320   c ,  320   d , and  320   e . Beginning with the first reference point  337 , the neckband first subsection  320   a  extends to and goes through both channels  940   a ,  944   a  of the first figure-8 slide  872   a , then transitions as the neckband second subsection  320   b  to and through both channels  940   b ,  944   b  of the second figure-8 slide  872   b , where direction is reversed such that the adjoining third neckband subsection  320   c  extends through both channels  944   a ,  940   a  of the first figure-8 slide  872   a . Next, the adjoining fourth neckband subsection  320   d , which extends away from the slide  872   a , reverses the neckband path for a second time to travel through both channels  944   b ,  940   b  of the second figure-8 slide  872   b , with the adjoining fifth neckband subsection  320   e  continuing to the second reference point  347 . 
       FIGS. 30A through 30D  are front views of the exemplary slides  872 ,  926 ,  928  and  930  for incorporation into an endless neckband length adjustment system to enable neckband length adjustment of endless and Mobius plumb neckties and bow ties, as well as ended plumb neckties and ended bowties.  FIGS. 30A and 30B  illustrate prior art slides  872  and  926 . Slide  872  is shown in  FIGS. 26 and 29  applied in a configuration according to the invention which adjusts the length of an endless portion of a neckband  120  or  320  or strap in an endless neckband or strap length adjustment system  900 .  FIGS. 30C and 30D  illustrate slides  928  and  930  according to embodiments of the invention. The figure-8 slide  872  of  FIG. 30A  and the figure-9 slide  926  of  FIG. 30B  may be incorporated into the exemplary endless neckband length adjustment system  900  shown in  FIGS. 26 and 29 . These slides may be passed onto the neckband or strap before the neckband or strap end regions are attached or connected to create an endless loop as described with reference to  FIG. 23 . The figure-Z slide  928  of  FIG. 30C  and the figure-M slide  930  of  FIG. 30D  are exemplary of open slides according to the invention. The slides  928  and  930  can be placed on the neckband  120  or, more generally, on a strap or a strip of material, before or after the end regions are attached to one another to create an endless loop as in  FIG. 23 , or attached to termination points such as attachment of a strap to carry a bag or a backpack. Each of the slides  872 ,  926 ,  928 , and  930  includes a first rib  938 , a first channel  940 , a second rib  942 , a second channel  944 , and a third rib  946 . The ribs may connect to form open spaces or closed spaces that further define the channels  940 ,  944 . The figure-8 slide  872  of  FIG. 30A  includes closed channels  940  and  944 . The figure-9 slide  926  of  FIG. 30B  includes the closed channel  944 , the first rib  938  ending in a first gaff  948 , and the open channel  940  which is open as a result of a gap  958  between the first gaff  948  and the second rib  942 . The gap  958  is of sufficient size to pass a neckband  120  or other strip of fabric into the channel  940 . 
     The figure-Z slide  928  of  FIG. 30C  also includes an open channel  940  with the rib  938  ending in a first gaff  948 , and further includes another open channel  944  with rib  946  ending in a second gaff  950 . The two gaffs  948  and  950  occupy opposing positions on different sides of the slide  928 . The center rib  942  is (i) attached to the first rib  938  at a first slide neck  952 , and (ii) attached to the third rib  946  at a second slide neck  954 . Open channel  940  of the slide  928  is open as a result of a gap  958  between the first gaff  948  and the second rib  942 , and open channel  944  of the slide  928  is open as a result of a gap  960  between the second gaff  950  and the second rib  942 . The gaps  958 ,  960  are of sufficient size to pass a neckband  120 ,  320  or other strap or strip of fabric into the channels  940 ,  944 . 
     The figure-M slide  930  of  FIG. 30D  includes a first open channel  940  with rib  938  ending in a first gaff  948 , and a second open channel  944  with rib  946  ending in a second gaff  950 . The two gaffs  948  and  950  occupy positions adjacent one another. Along one end of the slide, the center rib  942  is (i) attached to the first rib  938  at a first slide neck  952  and (ii) is adjacently attached to the third rib  946  at a third slide neck  956 . Open channels  940 ,  944  of the slide  930  are open as a result of a gap  962 , between rib  942  and gaffs  948  and  950 . The gap  962  is of sufficient size to pass a neckband  120 ,  320  or other strap or strip of fabric into the channels  940 ,  944 . 
     Although the illustrated slides may be referred to as hardware, composition of the slides is not limited to any particular metallic or nonmetallic material. The slides may, for example, be formed of metal or plastic. A slide may have resilient or spring-like qualities that facilitate bending to expand a gap adjoining a gaff. 
       FIG. 31A  is a perspective front view of another endless size adjustment system  900  according to an embodiment of the invention. While  FIGS. 26 and 29  illustrate the use of prior art figure-8 slides in the endless size adjustment system  900 ,  FIG. 31A  illustrates application of a figure-Z slide  928  and a figure-M slide  930  to an endless strap  924  according to embodiments of the invention. The slides  928 ,  930  in  FIG. 31B  lie generally in the same plane as the strap  924 . The term strap as used herein includes neckbands, belts, leashes, and other flat, relatively long, and relatively narrow strips, in addition to elements commonly referred to as straps including luggage straps, backpack straps, tie-down straps, and purse straps. An ended strap has at least one end which may be manipulated for adjustment. An endless strap does not have at least one end which may be manipulated for adjustment; an endless strap may result from attaching opposing ends of an ended strap to each other, or from attaching each opposing end of an ended strap to other components including fabric or hardware. 
     In the view of  FIG. 31B , the slides  928 ,  930  are spaced apart along a horizontal plane with the slides positioned in the plane. As can be seen in  FIG. 31B , the path of the strap  924  takes multiple turns, providing friction which effectively contributes to the stability of the length of the strap. By rotating the slides by 90° as shown in  FIG. 31C , the slides are generally perpendicular to the horizontal plane along which the slides are spaced apart. This renders the path of the strap through the horizontal channels  940 ,  944  more open to reduce the pressure on the strap so friction is reduced and movement of the strap  924  is less constrained. This rotated position is more conducive to changing the length of the neckband or strap. When the slides are turned as in  FIG. 31C , pulling the slides apart puts more of the neckband or strap between the slides. This reduces the effective length of the neckband or strap, i.e., it tightens the neckband. Pulling the neckband or strap so the slides move together puts less of the neckband or strap between the slides. This increases the effective length of the neckband or strap, e.g., it loosens the neckband about a neck. 
     Accordingly, features of embodiments of the endless size adjustment system  900  introduce novel topologies and novel constructions relative to providing a size adjustment mechanism for neck garb without requiring a break in the neckband, i.e., it is an endless size adjustment system. 
     Endless necktie embodiments of the endless size adjustment mechanism provide the advantages of eliminating complex connection hardware that must connect together or pierce fabric, thereby maintaining the integrity and elegance of the endless loop, and providing a size adjustment by the wearer based on real-time comfort. The endless size adjustment system  900  is also usable for size adjustability of other instances of straps such as clothing, webbing straps, belts, and tie-downs for transporting luggage, furniture, construction materials, and sports equipment. 
     Numerous embodiments of neckties have been described. According to one embodiment, an endless plumb necktie is tied in a manner similar to tying an overhand knot, resulting in the appearance of an ended neck tie with a blunt, non-pointed blade. Methods have been described for tying an endless plumb necktie in a simple manner that involves looping one projection through another, using unique features of the endless loop, this resulting in the appearance of an ended plumb neck tie with a blunt, non-pointed blade. 
     According to the invention, a Mobius plumb necktie, made from two pieces of fabric placed back-to-back. The necktie is tied in a simplified manner by looping one projection through another, using unique features of the endless loop, resulting in the appearance of an endless plumb neck tie having a blunt, non-pointed blade and with an outside knot showing both fabric pieces. This embodiment may be further shown to advantage by twisting the front blade to show both fabrics and securing the blade in place with a tie tack, tie bar, or other hardware. 
     An endless bow tie may be made from an endless loop by folding the endless loop at two places to form a left blade and a right blade, and tying the blades in an overhand knot next to the neck. This embodiment presents similar to an ended bow tie minus its bow tie tabs. An endless bow tie may also be configures from an endless loop by tying an overhand inside knot next to the neck followed by tying an outside knot thereby securing a left blade and a right blade. The inside knot and the outside knot together may be described as a single square knot. A variation on this embodiment adds a half rotation to the outside knot and, while less symmetric than the square knot, this variation mimics a way that ended bow ties are often tied. This embodiment presents similar to an ended bow tie minus its tabs. 
     A Mobius bow tie may also be made from a Mobius loop by folding the Mobius loop at two places to form a left blade and a right blade, and tying the blades in an overhand knot next to the neck. This embodiment presents similar to an ended bow tie minus its tabs. The characteristic Mobius twist may be shown to advantage on the outside knot or in one of the blades, or may be tucked out of sight. According to still another embodiment, a Mobius bow tie is configured from a Mobius loop by tying an overhand inside knot next to the neck followed by tying an outside knot which secures a left blade and a right blade. The inside knot and the outside knot together may be described as a single square knot. A variation on this embodiment adds a half rotation to the outside knot and, while less symmetric than the square knot, this variation mimics a way that ended bow ties are often tied. This embodiment presents similar to an ended bow tie minus its tabs; and the characteristic Mobius twist may be advantageously displayed on the outside knot or in one of the blades, or may be tucked out of sight. When a Mobius bow tie is formed from two fabrics and a twist is arranged diagonally across the outside knot, the two fabrics may be arranged to alternate from the first side of the neck band, to the first blade, to the first fabric of the outside knot, to the second fabric of the outside knot, to the second blade, and to the second side of the neck band. Variations on how the Mobius bow tie is “dressed” include constraining the Mobius twist to show on the outside knot or in one of the blades; or the Mobius twist may be tucked out of sight e.g., on the neckband below a collar. A variation on the endless bow tie embodiment involves doubling the length of the portion of the endless loop taken up by the left blade and the right blade, and dimpling this excess length and tucking it between the inside knot and the outside knot, the resulting double endless bow tie thus presenting similarly to an ended bow tie without tabs and with double blades. 
     According to yet embodiment, a double Mobius bow tie is made from a Mobius loop as described above, with double length included in the blade portion of the Mobius loop to allow for a double left blade and a double right blade, which double blades are dimpled and tucked between the inside and outside knots. The resulting double Mobius bow tie thus presents similar to an ended bow tie without tabs, with double blades, and with the characteristic Mobius twist. An embodiment of an asymmetric endless or Mobius bow tie is achieved by increasing or decreasing the length and/or width included in the blade portion of the endless loop or Mobius loop on one side while providing a different blade portion length and/or width on the other side. 
     Also according to the invention, a bow tie shaped ornament is created from an endless loop or Mobius loop. The ornamental bow tie may be adapted in size and choice of material to adorn various structures including bodies and fasteners such as pins, hairclips, bracelets, necklaces, and straps. Generally, embodiments of the invention may include enhancements in the shape of an endless or Mobius plumb necktie or bow tie by modifying the width of the endless loop or Mobius loop. For example, the region of the loop which becomes the presenting first blade may be narrower or wider than the region of the loop which becomes the second blade. Also, the shape of a double endless bow tie or double Mobius bow tie may be enhanced by modifying the width of the endless loop or Mobius loop. This may include a locally wide protrusion in the double-length loop, which protrusion may be tucked between the inside knot and outside knot to help anchor the tucked loop. 
     Generally, according to one series of embodiments, elongate lengths of material are formed into a tube, the ends of which are formed so as to allow for left twist, right twist, or no twist in completing steps comprising joining the ends of the tube to turn the tube into an endless or twisted or Mobius loop. Variations on this embodiment include but are not limited to finishing the endless or twisted or Mobius loop by stitching, bonding, gluing, or attaching appropriate fastener hardware. 
     Based on the varied embodiments described, the endless plumb neck tie  100 , the Mobius plumb neck tie  194 , the endless bow tie  200 , the Mobius bow tie  300 , the Mobius bow tie ornament  400 , the double endless bow tie  600 , and the double Mobius bow tie  604  may be made from woven fabric or ribbon including but not limited to silk, cotton, rayon, wool, linen, polyester, or blended fabric, or made from knitted fabric fashioned by cutting knit fabric, or by knitting, crocheting, tatting, or otherwise forming the custom loop shape, or by stamping, extruding, molding, or gluing suitable materials. However, the embodiments may consist of any other material that may be knotted without fracturing, including but not limited to leather, vinyl, nylon, paper, plastic, or plasticized materials. In one embodiment of a Mobius necktie, made from a fabric tube having contrasting fabrics back-to-back, color contrast between two such fabrics enhances the distinctive look of the Mobius twist. 
     The above described embodiments of the present invention are merely exemplary and descriptive of the principles. The present invention is not limited to these embodiments which may be amended or modified without departing from the scope of the present invention, which includes structural and functional equivalents. For example, the neck band may be adjustable to the size of the wearer&#39;s neck by use of other adjustable hardware arrangements than those illustrated in the figures. A rotatable hardware arrangement may be included to add or delete twists consisting of integer multiples of 180 degrees. Breakaway hardware in the neckband may be used for safety concerns as for the endless plumb neck tie where the lowest region of the blade could catch on a nearby structure. The scope of the present invention instead shall only be limited by the claims which follow.