On composite twisted unknots

Following Mathieu, Motegi and others, we consider the class of possible composite twisted unknots as well as pairs of composite knots related by twisting. At most one composite knot can arise from a particular V-twisting of an unknot; moreover a twisting of the unknot cannot be composite if we have applied more than a single full twist. A pair of composite knots can be related through at most one full twist for a particular V-twisting, or one summand was unaffected by the twist, or the knots were the right and left handed granny knots. Finally a conjectured characterization of all composite twisted unknots that do arise is given. Following Mathieu [Ma], Motegi [Mo] and others, we consider the class of possible composite twisted unknots. In essence, a twisting of a given knot K is constructed by pulling several strands away from the knot, momentarily cutting them, giving them a few full twists, and rejoining the strands to each other as they were before, as illustrated in Figure 1. A knot is composite if it can be described as two simpler knots spliced together. Motegi conjectured, and we show in Section 3, that a twisting of the unknot cannot be composite if we have given the unknot more than a single twist. In Section 2, we also describe all known examples of composite knots that do arise from a single twist of the unknot, and conjecture the list is complete. But once we have chosen our strands to twist, can we obtain two composite twisted unknots, the first from twisting once one way, the other from twisting once in the other direction? In Section 4, we show that a pair of composite knots can be related through at most a single full twist, or the twist affected only one summand, or that the knots were the right and left handed granny knots, which cannot be twisted unknots by [MY]. The pair of composite knots arising from the same twisting of an unknot would be twistings of one another, related through two full twists, an impossibility, since the only such knots cannot be twisted unknots. Finally we show every composite knot is related to an infinite number of other composite knots through some full twist. Let us formalize our terms. Let K be a knot in the interior of a standardly embedded solid torus V in S3 (Figure 1). Let A, pu c aV be the standard longitude, meridian of V in S3. For a simple closed curve v C aV such that [v] = [A] + \[/4t] in Hl(V Z), let V(s) = V(v) = VU (solid torus such that meridian lies on v) S3. We assume that the minimal geometric intersection of K with a meridinal disk of V is at least two. KA = K c V(ZA) then depends on K in V and i\ E Z. Generally we assume i\ > 0, by our choice of orientation of ,t. If K is unknotted in V(O) -S3, we call KA a twisted unknot. If K is a composite knot in V(O) we call KA a twisted composite knot. In general we consider KA to be a (A, V)-twisting Received by the editors August 8, 1994 and, in revised form, September 28, 1995. 1991 Mathematics Subject Classification. Primary 57M25. (?1997 American Mathematical Society 4429 This content downloaded from 207.46.13.85 on Sat, 28 May 2016 06:36:00 UTC All use subject to http://about.jstor.org/terms 4430 CHAIM GOODMAN-STRAUSS FIGURE 1. A twisting of a knot of K. We occasionally refer to a V-twisting, a (A, V)-twisting for any A\. Mathieu [Ma] described this twisting operation. A knot K is composite if a splitting sphere can separate S3 into balls each containing a single, knotted arc of K, a knot summand. In our analysis of composite twisted unknots that arise when A = 1, there are two cases: either there is a sphere splitting the knot summands that is punctured twice by OV or every splitting sphere must be punctured at least four times. With Theorem 2.2 we classify all composite twisted unknots arising from the former. We conjecture that the latter cannot occur. Motegi [Mo] recently conjectured that each presentation of K in V gives rise to at most one composite twisted unknot and that this Ka arises from a single twist of K. In [Mo] Motegi proves any such K,\ has A\ 3, V)-twisting of a composite knot cannot be composite. Miyazaki and Yasuhara give a sufficient condition for a knot's not being a twisted unknot in [MY] and produce many examples of composite knots that meet this condition. The techniques of Section 2 are explicit, and are independent of those of Sections 3 and 4. Our proofs of Theorems 3.1 and 4.1 will rely heavily on the techniques developed in [GLI] and expanded upon in [GL2]. We will assume familiarity with those papers. Loosely speaking, these techniques apply to punctured low genus surfaces, properly embedded in a manifold with torus boundary. If the conditions are right, the techniques produce a negative result, that the surfaces could not have been embedded as desired. Although the lemmas and theorems in [GLI], [GL2], [CGLS] etc. arise from the graphs of intersection of two punctured spheres, we can carefully adapt them to the graphs of intersection of other punctured surfaces. For the proof of Theorem 4.1 we must also introduce loop types to produce ring graphs which give rise to the exceptional case of the granny knots related by 2-twisting. The author gratefully acknowledges John Luecke's patient and generous guidance. This content downloaded from 207.46.13.85 on Sat, 28 May 2016 06:36:00 UTC All use subject to http://about.jstor.org/terms ON COMPOSITE TWISTED UNKNOTS 4431 1. THE GEOMETRIC SETTING We will first examine a punctured disk arising from the disk bounding an unknot K in V(O), and a punctured sphere derived from a splitting two-sphere for composite K in V(A). The techniques of [GL1] and [GL2] describe the graphs of intersection of such surfaces, the complex formed by their union, and the nature of any manifold that might contain such a complex. As we will see in Section 3, if A > 2, such a complex cannot be correctly placed in S3, for a contradiction. Take V in V(O) = S3, the complement of a small open neighborhood of an unknot H. Thus for any integer A, V(A) = H(1/A), the 1/A Dehn-surgery on H. Let D* be an embedded disk in V(O) with AD* = K, such that D* intersects H transversally and minimally. Let D = D*int(N(H)) = D* n V, a punctured disk. Assume KA is composite. Then let A* be an embedded two-sphere in V(A) separating K into two knotted arcs, such that A* intersects H transversally and minimally. Let A = A* -int(N(H)) = A* nfV, a punctured sphere. We can choose A* so that A and D meet minimally and transversally in V. There are three kinds of curves of intersection of A and D in V: arcs with both ends on aV; simple closed curves of intersection; and either two arcs a1, a2 c AnD with one end on aV, the other end in A n K, or one arc ao C A n D with both ends in AnK. Note that a simple closed curve of intersection s cannot bound a disk in either A or D. If s bounds a disk in both A and D then A and D do not intersect minimally. If s bounds a disk in A but not D, then D does not intersect H minimally. If s bounds a disk in D but not A, A does not intersect D or H minimally. We will consider the graphs of only the arcs of intersection of A and D and will take note of the simple closed curves of intersection only in our final geometric arguments. Let GA, [GD] be this graph on A, [D]. A pair of these graphs is depicted in Figure 2. Their geometric interpretation may be clearer after examining the constructions in Section 2. We note that on aV, the components of AA, [9D] are parallel and thus share the well-defined slope 1/A, [0]. Moreover, each component of AA meets each component of AD in exactly A points. Thus GA and GD can be described by the nomenclature of pp. 386-387 [GL1].