Kimihiko Motegi

Only single twists on unknots can produce composite knots

Let K be a knot in the 3-sphere S3, and D a disc in S3 meeting K transversely more than once in the interior. For non-triviality we assume that IK n DI > 2 over all isotopy of K. Let Kn (C S3) be a knot obtained from K by cutting and n-twisting along the disc D (or equivalently, performing 1/n-Dehn surgery on OD). Then we prove the following: (1) If K is a trivial knot and Kn is a composite knot, then Inj < 1; (2) if K is a composite knot without locally knotted arc in S3 -,OD and Kn is also a composite knot, then Inj < 2. We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.

Seifert-fibered surgeries which do not arise from primitive/Seifert-fibered constructions

We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Deans primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert-fibered surgeries arise from Deans primitive/Seifert-fibered construction. The (-3, 3,5)-pretzel knot belongs to both of the infinite families.

DEHN SURGERY ON KNOTS IN SOLID TORI CREATING ESSENTIAL ANNULI

Let M be a 3-manifold obtained by performing a Dehn surgery on a knot in a solid torus. In the present paper we study when M contains a separating essential annulus. It is shown that M does not contain such an annulus in the majority of cases. As a corollary, we prove that symmetric knots in the 3-sphere which are not periodic knots of period 2 satisfy the cabling conjecture. This is an improvement of a result of Luft and Zhang. We have one more application to a problem on Dehn surgeries on knots producing a Seifert fibred manifold over the 2-sphere with exactly three exceptional fibres.

L-space surgery and twisting operation

A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, i.e. a rational homology 3-sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c and twist K n times along c to obtain a twist family { K_n }. We give a sufficient condition for { K_n } to contain infinitely many L-space knots. As an application we show that for each torus knot and each hyperbolic Berge knot K, we can take c so that the twist family { K_n } contains infinitely many hyperbolic L-space knots. We also demonstrate that there is a twist family of hyperbolic L-space knots each member of which has tunnel number greater than one.

Toroidal and annular Dehn surgeries of solid tori

Abstract We obtain an infinite family of hyperbolic knots in a solid torus which admit half-integral, toroidal and annular surgeries. Among this family we find a knot with two toroidal and annular surgeries; one is integral and the other is half-integral, and their distance is 5. This example realizes the maximal distance between annular surgery slopes and toroidal ones, and that between annular surgery slopes.

Haken manifolds and representations of their fundamental groups in SL(2, C)

Abstract It is shown in [2] that if the fundamental group of a compact orientable irreducible 3-manifold M has a positive-dimensional SL(2, C )-character variety, then M is a Haken manifold. We show however that the converse is not true. That is there exist infinitely many Haken manifolds whose fundamental groups have a finite number of representations in SL(2, C ) up to equivalence. In particular, they have 0-dimensional SL(2, C )-character varieties.

Seifert fibering surgery on periodic knots

Abstract We show that if a periodic knot K in the 3-sphere yields a Seifert fibered manifold by Dehn surgery, then the quotient of K by the group action generated by any periodic map of K is a torus knot, except for a special case. We also consider what Seifert fibered manifolds are obtained by Dehn surgery on periodic knots. If a non-torus, periodic knot yields a Seifert fibered manifold M , then the base space of M is the 2-sphere; and some pair of exceptional fibers in M has indices coprime provided that M contains at most three exceptional fibers.

On satellite knots

Throughout this paper we work in the smooth category and assume that all knots are oriented and consider two knots K 1 and K 2 to be equivalent if and only if there is an orientation preserving homeomorphism h : S 3 → S 3 which carries K 1 onto K 2 so that their orientations match. We write K 1 ≅ K 2 if K 1 and K 2 are equivalent and – K denotes the knot obtained from K by inverting its orientation.

Primeness of twisted knots

Let V be a standardly embedded solid torus in S3 with a meridianpreferred longitude pair (,u, A) and K a knot contained in V. We assume that K is unknotted in S3 . Let f,n be an orientation-preserving homeomorphism of V which sends A to A + n,u . Then we get a twisted knot Kn = fn(K) in S3 Primeness of twisted knots is discussed and we prove: A twisted knot Kn is prime if InI > 5. Moreover, {K },nEZ contains at most five composite knots.

Networking Seifert surgeries on knots, III

How do Seifert surgeries on hyperbolic knots arise from those on torus knots? We approach this question from a networking viewpoint introduced in [9]. The Seifert surgery network is a 1‐dimensional complex whose vertices correspond to Seifert surgeries; two vertices are connected by an edge if one Seifert surgery is obtained from the other by a single twist along a trivial knot called a seiferter or along an annulus cobounded by seiferters. Successive twists along a “hyperbolic seiferter” or a “hyperbolic annular pair” produce infinitely many Seifert surgeries on hyperbolic knots. In this paper, we investigate Seifert surgeries on torus knots that have hyperbolic seiferters or hyperbolic annular pairs, and obtain results suggesting that such surgeries are restricted. 57M25; 57M50, 57N10 Dedicated to Sadayoshi Kojima on the occasion of his 60 th birthday