# List of knot theory topics

Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

## Knots, links, braids

* Knot (mathematics) gives a general introduction to the concept of a knot.
** Two classes of knots: torus knots and pretzel knots
** Cinquefoil knot also known as a (5, 2) torus knot.
** Figure-eight knot (mathematics) the only 4-crossing knot
** Granny knot (mathematics) and Square knot (mathematics) are a connected sum of two Trefoil knots
** Perko pair, two entries in a knot table that were later shown to be identical.
** Stevedore knot (mathematics), a prime knot with crossing number 6
** Three-twist knot is the twist knot with three-half twists, also known as the 5$_2$ knot.
** Trefoil knot A knot with crossing number 3
** Unknot
* Knot complement, a compact 3 manifold obtained by removing an open neighborhood of a proper embedding of a tame knot from the 3-sphere.
Notation used in knot theory:
* Conway notation
* Dowker–Thistlethwaite notation (DT notation)
* Gauss code (see also Gauss diagrams)
* continued fraction regular form

### General knot types

* 2-bridge knot
* Alternating knot; a knot that can be represented by an alternating diagram (i.e. the crossing alternate over and under as one traverses the knot).
* Berge knot a class of knots related to Lens space surgeries and defined in terms of their properties with respect to a genus 2 Heegaard surface.
* Cable knot, see Satellite knot
* Chiral knot is knot which is not equivalent to its mirror image.
* Double torus knot, a knot that can be embedded in a double torus (a genus 2 surface).
* Fibered knot
* Framed knot
* Invertible knot
* Prime knot
* Legendrian knot are knots embedded in $\mathbb {R} ^{3}$ tangent to the standard contact structure.
* Lissajous knot
* Ribbon knot
* Satellite knot
* Slice knot
* Torus knot
* Transverse knot
* Twist knot
* Virtual knot
* welded knot
* Wild knot

### Links

* Borromean rings, the simplest Brunnian link
* Brunnian link, a set of links which become trivial if one loop is removed
* Hopf link, the simplest non-trivial link
* Solomon's knot, a two-ring link with four crossings.
* Whitehead link, a twisted loop linked with an untwisted loop.
* Unlink
General types of links:
* Algebraic link
* Hyperbolic link
* Pretzel link
* Split link
* String link

### Tangles

* Tangle (mathematics)
* Algebraic tangle
* Tangle diagram
* Tangle product
* Tangle rotation
* Tangle sum
* Inverse of a tangle
* Rational tangle
* Tangle denominator closure
* Tangle numerator closure
* Reciprocal tangle

### Braids

* Braid theory
** Braid group

## Operations

* Band sum
* Flype
* Fox n-coloring
** Tricolorability
* Knot sum
* Reidemeister move

### Elementary treatment using polygonal curves

* elementary move (R1 move, R2 move, R3 move)
* R-equivalent
* delta-equivalent

## Invariants and properties

* Knot invariant is an invariant defined on knots which is invariant under ambient isotopies of the knot.
* Finite type invariant is a knot invariant that can be extended to an invariant of certain singular knots
* Knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.
** Alexander polynomial and the associated Alexander matrix; The first knot polynomial (1923). Sometimes called the Alexander–Conway polynomial
** Bracket polynomial is a polynomial invariant of framed links. Related to the Jones polynomial. Also known as the Kauffman bracket.
** Conway polynomial uses Skein relations.
** Homfly polynomial or HOMFLYPT polynomial.
** Jones polynomial assigns a Laurent polynomial in the variable t$^{1/2}$ to the knot or link.
** Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman.
* Arf invariant of a knot
* Average crossing number
* Bridge number
* Crosscap number
* Crossing number
* Hyperbolic volume
* Kontsevich invariant
* Linking number
* Milnor invariants
* Racks and quandles and Biquandle
* Ropelength
* Seifert surface
* Self-linking number
* Signature of a knot
* Skein relation
* Slice genus
* Tunnel number, the number of arcs that must be added to make the knot complement a handlebody
* Writhe

## Mathematical problems

* Berge conjecture
* Birman–Wenzl algebra
* Clasper (mathematics)
* Eilenberg–Mazur swindle
* Fáry–Milnor theorem
* Gordon–Luecke theorem
* Khovanov homology
* Knot group
* Knot tabulation
* Knotless embedding
* Linkless embedding
* Link concordance
* Link group
* Link (knot theory)
* Milnor conjecture (topology)
* Milnor map
* Möbius energy
* Mutation (knot theory)
* Physical knot theory
* Planar algebra
* Smith conjecture
* Tait conjectures
* Temperley–Lieb algebra
* Thurston–Bennequin number
* Tricolorability
* Unknotting number
* Unknotting problem
* Volume conjecture

### Theorems

* Schubert's theorem
* Conway's theorem
* Alexander's theorem

## Lists

* List of mathematical knots and links
** List of prime knots