Mikami Hirasawa

THE GORDIAN COMPLEX OF KNOTS

In this paper, we define the Gordian complex of knots, which is a simplicial complex whose vertices consist of all oriented knot types in the 3-sphere. We show that for any knot K, there exists an infinite family of distinct knots containing K such that any pair (Ki, Kj) of the member of the family, the Gordian distance dG(Ki, Kj) = 1.

Dehn surgeries on strongly invertible knots which yield lens spaces

In this article we show no Dehn surgery on nontrivial strongly invertible knots can yield the lens space L(2p, 1) for any integer p. In order to do that, we determine band attaches to (2, 2p)-torus links producing the trivial knot.

SEIFERT SURFACES IN OPEN BOOKS, AND A NEW CODING ALGORITHM FOR LINKS

We introduce a new standard form of a Seifert surface F. In that standard form, F is obtained by successively plumbing flat annuli to a disk D, where the gluing regions are all in D. We show that any link has a Seifert surface in the standard form, and thereby present a new way of coding a link. We present an algorithm to read the code directly from a braid presentation.

Twisted Alexander polynomials of 2-bridge knots associated to dihedral representations

Let p be an odd prime and D_p a dihedral group of order 2p. Let \rho: G(K) --> D_p --> GL(p,Z) be a non-abelian representation of the knot group G(K) of a knot K in 3-sphere. Let \Delta_{\rho,K} (t) be the twisted Alexander polynomial of K associated to \rho. Let H(p) is the set of 2-bridge knots K, such that G(K) is mapped onto a non-trivial free product Z/2 * Z/p. Then we prove that for any 2-bridge knot K in H(p), \Delta_{\rho,K}(t) is of the form \Delta_{K}(t)/(1-t) f(t) f(-t) for some integer polynomial f(t), where \Delta_K (t) is the Alexander polynomial of K. Further, it is proved that f(t) \equiv {\Delta_K (t)/(1+t)}^n (mod p). Later we discuss the twisted Alexander polynomial associated to the general metacyclic representation.

Triviality and splittability of special almost alternating links via canonical Seifert surfaces

Abstract In this article we give a sufficient condition for almost alternating diagrams to represent a non-trivial knot or a non-splittable link. Using this result, we determine which special almost alternating diagrams represent the unknot or a splittable link.

BRIDGE PRESENTATIONS OF VIRTUAL KNOTS

We study bridge presentations of virtual knots, and determine the virtual bridge numbers of pseudo-prime virtual knots with real crossing numbers less than 5, except two virtual knots.

Fibred double torus knots which are band-sums of torus knots

A double torus knot K is a knot embedded in a Heegaard surface H of genus 2, and K is non-separating if H K is connected. In this paper, we determine the genus of a non-separating double torus knot that is a band-connected sum of two torus knots. We build a bridge between an algebraic condition and a geometric requirement (Theorem 5.5), and prove that such a knot is fibre d if (and only if) its Alexander polynomial is monic, i.e. the leading coeffici ent is 1. We actually construct fibre surfaces, using T. Kobayashi’s geometric ch aracterization of a fibred knot in our family. Separating double torus knots are also discussed in the last section.

Pre-taut sutured manifolds and essential laminations

In 1989, D. Gabai and U. Oertel [8] introduced the concept of the essential lamination, which is a hybrid object lying between incompressible surfaces and taut foliations, and generalizing both. We say that a 3-manifold is laminar if it contains an essential lamination. An important result of [8] is that the universal covers of laminar manifolds are homeomorphic to R. This fact furnishes a strong method for studying the manifolds obtained by Dehn surgery along knots, especially concerning Property P Conjecture (nontrivial Dehn surgery on a nontrivial knot in 3 never yields a simply-connected manifold) and Cabling Conjecture (Dehn surgery on a non-cable knot cannot yield a reducible manifold). For example, see [4] for non-torus alternating knots, [3], [12] for 2-bridge knots, [17] for most algebraic knots and [9] for knots with some kind of essential tangle decompositions. We note that by [8] a 3-manifold is laminar if and only if it contains an essential branched surface (for the definition see §2), and the above authors who followed [8] obtained their results by constructing essential branched surfaces. We note that sutured manifold theory was used in [14] and [18]. One of their approaches is to construct a closed essential branched surface in the exterior ( ) of a knot and show that remains essential after any nontrivial Dehn filling along ∂ ( ) (we call such persistently essential). Then we see, by [8], that has Property P in a strong form and that the cabling conjecture is true for . (We say that a knot has strong Property P if every manifold obtained by a nontrivial Dehn surgery along has universal cover R.) It is, however, an open question whether or not every knot with strong Property P admits a persistently essential lamination in its complement. In [1], [2], M. Brittenham had a paradigm shift in proving strong Property P for knots. Instead of constructing a branched surface in the complement of a given knot, he first constructed a branched surface and then embedded a knot in its complement. More precisely, he first constructed a closed branched surface in 3 from any in-

FIBERED TORTI-RATIONAL KNOTS

A torti-rational knot, denoted by K(2a,b|r), is a knot obtained from the 2-bridge link B(2a,b) by applying Dehn twists an arbitrary number of times, r, along one component of B(2a,b). We determine the genus of K(2a,b|r) and solve a question of when K(2a,b|r) is fibred. In most cases, the Alexander polynomials determine the genus and fibredness of these knots. We develop both algebraic and geometric techniques to describe the genus and fibredness by means of continued fraction expansions of b/2a. Then, we explicitly construct minimal genus Seifert surfaces. As an application, we solve the same question for the satellite knots of tunnel number one.