Intelligence of Low Dimensional Topology 2006

PREFACE.

ORGANIZING COMMITTEES.

CONTENTS.

The combinatorial structures of the Ford domains of quasifuchsian punctured torus groups are characterized by T. Jorgensen. In this paper, we try to find an analogue of Jorgensen's theory for a certain manifold with a pair of once-punctured tori as boundary. This is a report on a work in progress.

We present a symbolic computations for developing cohomology theories of algebraic systems. The method is applied to coalgebra self-distributive maps to recover low dimensional differential maps.

This paper is concerned with the Khovanov homology of links which admit a semi-free ℤ/pℤ-symmetry. We prove that if we consider Khovanov homology with coefficients in the field 𝔽₂, then the ℤ/pℤ-symmetry (for any odd integer p) of the link extends to an action of the group ℤ/pℤ on the Khovanov homology.

Self C_k-equivalence is a natural generalization of Milnor's link homotopy. It has been long known that a Milnor invariant with no repeated index, for example μ(123), is a link homotopy invariant. We show that Milnor numbers with repeated indices are invariant under self C_k-moves, and apply these invariants to study links up to self C_k-equivalence. Using these techniques, we are able to give a complete classification of 2-string links up to self delta-equivalence.

We present some results on estimates of the Morse-Novikov numbers for knots and links. Using these, we show the Morse-Novikov numbers concretely for some knots and links.

We show that minimal genus Seifert surfaces for 2-bridge links can be isotoped to braided Seifert surfaces on minimal string braids. In particular, we can obtain a minimal string braid for a given 2-bridge link, where the word length of the braid thus obtained is shortest, among all braid presentations, in terms of the band generators of braids.

The Alexander–Conway polynomial is reconstructed in a manner similar to the way the Jones polynomial is constructed by using the Kauffman bracket polynomial. This is a summary of the reconstruction.

In this paper, we give a formula for the quandle cocycle invariants associated with the Fox p-coloring of torus links. In case torus knots this formula was given by S. Asami and S. Satoh.

This article explains how the authors obtained the list of prime knots of arc index not bigger than 10.

In this paper we define the p-adic framed braid group , arising as the inverse limit of the modular framed braids. An element in can be interpreted geometrically as an infinite framed cabling. contains the classical framed braid group as a dense subgroup. This leads to a set of topological generators for and to approximations for the p-adic framed braids. We also construct a p-adic Yokonuma-Hecke algebra Y_∞,n(u) as the inverse limit of the classical Yokonuma-Hecke algebras. These are quotients of the modular framed braid groups over a quadratic relation. Finally, we construct on this new algebra a p-adic linear trace that supports the Markov property. Paper presented at the 1017 AMS Meeting.

We investigate whether a p/q-surgery (p ≥ 2) along a knot K in a homology 3-sphere is a Seifert fibered space or not by using Reidemeister torsion. We obtain some necessary conditions about values of the Alexander polynomial for K yielding a Seifert fibered space at root of unities. By using the conditions, we prove that if a p/q-surgery along a knot K whose Alexander polynomial is Δ_K(t) = t² − 3t + 1 is a Seifert fibered space, then we have p = 2 or 3.

Miyazawa polynomials are invariants of virtual links. We discuss some features of Miyazawa polynomials and give a table of virtual knots whose real crossing numbers are equal or less than four. Furthermore we determine virtual crossing numbers of some knots in the table.

Quandles and their homologies are used to construct invariants of oriented links or oriented surface-links in 4-space. On the other hand the knot quandle can still be defined in the case where the links or surface-links are not oriented, but in this case it cannot be used to construct homological invariants. Here we introduce the notion of a quandle with a good involution, and its homology groups. We can use them to construct invariants of unoriented links and unoriented, or non-orientable, surface-links in 4-space.

We give a formula expressing any finite type invariant for oriented two-component links of order ≤ 4 in terms of the linking number, the Conway and HOMFLYPT polynomials.

We consider the linear representation of the mapping class group of the 4–punctured sphere which arises from the Iwahori–Hecke algebra representation corresponding to the Young diagram [2, 2]. We determine its kernel and show that the representation could be considered as the Jones representation of genus 1, which turns out to be a faithful representation of PSL(2, ℤ).

This paper is a summary of research on the relationship between quantum topology and quantum information theory.

In this paper we sketch our proof of a Markov Theorem for the virtual braid group using L-move techniques.

We examine the volume conjecture for the HOMFLY polynomial instead of the Jones polynomial. For the figure-eight knot, we explicitly obtain the limits of the colored HOMFLY polynomial.

We discuss a conjecture of Jones on braid index and algebraic crossing number. We deform it to a stronger conjecture and show many evidences and some ways to approach the conjectures.

The set of the fundamental groups of n-dimensional manifold-links in Sⁿ⁺² for n > 2 is equal to the set of the fundamental groups of surface-links in S⁴. We consider the subset of this set consisting of the fundamental groups of r-component, total genus g surface-links with H₂(G) ≅ H. We show that the set is a non-empty proper subset of for every integer g ≥ 0 and every abelian group H generated by 2g elements. We also determine the set to which the fundamental group of every classical link belongs, and investigate the set to which the fundamental group of every virtual link belongs.

A well-order was introduced on the set of links by A. Kawauchi [3]. This well-order also naturally induces a well-order on the set of prime link exteriors and eventually induces a well-order on the set of closed connected orientable 3-manifolds. With respect to this order, we enumerated the prime links with lengths up to 10 and the prime link exteriors with lengths up to 9. Our present plan is to enumerate the 3–manifolds by using the enumeration of the prime link exteriors. In this paper, we show our latest result in this plan and as an application, give a new proof of the identification of Perko's pair.

A partial order on the set of the prime knots can be defined by existence of a surjective homomorphism between knot groups. We determined the partial order for all knots of Rolfsen's knot table in the previous paper [3]. In this paper, we show a sketch of the proof and consider further problems which are caused by the above result.

We express the volume of a simplex in spherical or hyperbolic spece by iterated integrals of differential forms following Schläfli and Aomoto. We study analytic properties of the volume function and describe the differential equation satisfied by this function.

In this article, we introduce a method to construct invariants of the stably equivalent surface links in ℝ⁴ by using invariants of classical knots and links in ℝ³. We give invariants derived from this construction with the Kauffman bracket polynomial.

We use surgery along Brunnian links to relate, via a certain isomorphism, the Goussarov-Vassiliev theory for Brunnian links and the finite type invariants of integral homology spheres. To do so, we show that no finite type invariant of degree < 2n − 2 can vary under surgery along an (n + 1)-component Brunnian link in a compact connected oriented 3-manifolds, where the framing of the components is in .

As an extension of the Yamada polynomial for spatial graphs, we construct a polynomial invariant for virtual graphs. Since a virtual link can be regarded as a virtual graph having no vertices, the polynomial is an invariant for virtual links. We show that the invariant is useful for detecting non-classicality of a virtual knot.

In this note, we would like to discuss leisurely analogies between knot theory and number theory, focused on the variation of representations of the knot and prime groups. More precisely, we will discuss two things: First, we recall

1. the basic analogies between knots and primes and, based on them, secondly we will discuss some analogies between

2. the deformations of hyperbolic structures on a knot complement and of modular Galois representations

The latter analogy was first pointed out by Kazuhiro Fujiwara ([6]. See also [14]). We will make his observation more precise and discuss intriguing analogous features between SL₂(ℂ) Chern-Simons invariants and p-adic modular L-functions for GL₂.

For finite depth foliations, we define an invariant called “gap” which describes the maximal “gap” of the depths of adjacent leaves. Then for a certain 3-manifolds, we give an estimation of the depth of the given foliation by using gap.

We present the complex analytic and principal complex analytic realizability of a link in a 3-manifold M as a tool for understanding the complex structures on the cone C(M).

In this short article we report that for any odd integer m there exist an achiral spatial complete graph on 5 vertices and an achiral spatial complete bipartite graph on 3 + 3 vertices whose Simon invariants are equal to m.

A spider is a robot with n arms such that each arm is of length 1 + 1 and has a rotational joint in the middle, and that the endpoint of the kth arm is fixed to . We assume that it can move only in a plane. The configuration space of such planar spiders is studied. It is generically diffeomorphic to a connected orientable closed surface.

Invariants of 3-manifolds with 1-dimensional cohomology classes were introduced by Turaev-Viro, and developed by Gilmer for 3-manifolds obtained by 0-surgery along knots. They formulated the invariants as an equivariant version of quantum invariants for infinite cyclic covers of the 3-manifolds using TQFT functors of the quantum invariants. In this article, we give a survey on a construction of the invariants based on surgery presentations of knots.

We give a geometric description of welded links in the spirit of Kuperberg's description of virtual links: “What is a virtual link?” [8].

The set of homology cobordisms of a surface has a natural monoid structure. We give an application of higher-order Alexander invariants, which originated with Cochran and Harvey, to this monoid by using its Magnus representation and Reidemeister torsions.

By using Markoff (trace) maps, we give an alternative proof to a main result of the author's joint paper with Tomotada Ohtsuki and Robert Riley [8], which gives a systematic construction of epimorphisms between 2-bridge link groups.

A diagram of a surface-knot consists of a disjoint union of compact conneted surfaces. The sheet number of a surface-knot is the minimal number of such connected surfaces among all possible diagrams of the surface-knot. This is a generalization of the crossing number of a classical knot. We give a lower bound of the sheet number by using quandle-colorings of a diagram and the cocycle invariant of a surface-knot.

We showed the following in [8]: for any knot (i.e. one-component link) represented as a closed n-braid (n ≥ 3), there exists an infinite sequence of pairwise non-conjugate (n + 1)-braids representing the knot. Using the same technique, we construct an infinite sequence of pairwise non-conjugate 4-braids representing the same knot of braid index 4. Consequently, we have that M. Hirasawa's candidates are actually such infinite sequences.

This is an exposition of our work on tabulating mutants, and the related examination of various (mostly polynomial) invariants, in particular the colored Jones polynomial.

A chart is an oriented labelled graph in a 2-disk and C-moves are moves for charts which consist of three classes of moves: a CI-, CII- and CIII-move. Both CII- and CIII-move are local, but a CI-move is global. Carter and Saito proved a basic result about CI-moves which says that any CI-move can be realized by a finite sequence of seven types of local moves, but it seems that there are some ambiguous arguments in their proof. The purpose of this note is to give an outline of a precise proof for their assertion by a different approach.

We study when a lens space is homeomorphic to p-Dehn surgery of a knot in S³. Using correction term defined by P. Ozsváth and Z. Szabó, we will prove some obstructions of Alexander polynomial of K.

Let M be a rational homology 3-sphere. Let K be a null-homologus knot in M. We compute the Casson-Walker invariant of the cyclic covering space of M branched over K. Using C. Lescop's formula, we calculate the Casson-Walker invariant.

It is shown that for a positive integer g, there is a free genus one knot which admits g + 1 mutually disjoint and mutually non-equivalent incompressible Seifert surfaces each of genus g. As an application we show that for a positive integer g, the genus two handlebody contains g + 1 mutually disjoint and mutually non-isotopic separating incompressible surfaces each of genus g.

For any graph G we define bigraded cohomology groups whose graded Euler characteristic is a multiple of the Yamada polynomial of G.

In this note, we give an idea of the proof of a conjecture due to J. Dubois and R. Kashaev [6]. It says that there exists a relationship between: (i) differential coefficients of the Milnor torsion of the complement of a knot and (ii) limit values of the non-abelian twisted Reidemeister torsion at bifurcation points of the SL (2, ℂ)-representation variety of the knot group.

We study open books (or open book decompositions) of a closed oriented 3-manifold which support overtwisted contact structures. We focus on a simple closed curve along which one can perform Stallings twist, called “twisting loop”.

Let X be a compact connected orientable Haken 3-manifold with boundary, and let M(X) denote the 4-manifold ∂(X × D²). We show that if (f, b) : N → M(X) is a degree 1 TOP normal map with trivial surgery obstruction in L₄(π₁(M(X))), then (f, b) is TOP normally bordant to a homotopy equivalence f′ : N′ → M(X). Furthermore, for any CW-spine B of X, we have a UV¹-map p : M(X) → B and, for any ɛ > 0, f′ can be chosen to be a p⁻¹(ɛ)-homotopy equivalence.

For a surface diagram, there is a cell-complex associated with the surface diagram. The index of a surface diagram is the rank of the second homology group of the cell-complex. We present that if a 2-knot has the triple point number four, its t-minimal surface diagram has four branch points and its cell-complex has index 1, then the knot group of the surface-knot for the surface diagram has a presentation of the knot group of the double twist spun trefoil.

Harer, Kas and Kirby conjectured that every handle decomposition of the Dolgachev surface E(1)_2,3 requires both 1- and 3-handles. In this article, we construct a smooth 4-manifold which has the same Seiberg-Witten invariant as E(1)_2,3 and has neither 1- nor 3-handles in a handle decomposition.