Journal of Knot Theory and Its Ramifications

The purpose of this note is to give a formula for the Casson knot invariant (the second coefficient of the Conway polynomial) of a 2-bridge knot, calculating without using a continued fraction expansion.

We prove that the knot invariants given by quandles can be recovered by invariants given by traces on braided vector spaces, i.e. quantum invariants.

In this paper, we treat C-complexes for links defined by D. Cooper, and show the Component-isotopy Lemma for arbitrary links, which is a generalization of Cooper's result for 2-component links. By applying the technique in the present paper, we show a similar result for knots, which is called Surface-homotopy Lemma.

A differential geometric approach to Milnor-Massey higher order linking numbers for generic links is devised, via Chen's theory of iterated path integrals. Massey linking numbers arise from curvature forms of nilpotent "topological" connections, determined by the link structure, and interpreted in terms of intersection theory, leading to a fairly easy computation thereof. A version of the Turaev-Porter theorem expressing equality of Milnor and Massey linking numbers is also exhibited along the same lines, by computing suitable flat connection parallel transport operators in two different ways.

Using the notion of surgery on objects called Y-graphs and claspers by Goussarov and Habiro, one can define a theory of finite type invariants of closed 3-manifolds. The paper discusses upper bounds for the number of invariants, and focuses on two surprises that arise: One surprise is that the upper bounds depend on a bit more than a choice of generators for H₁. A complementary surprise a curious brane relation (in two flavors, open and closed) which shows that the upper bounds are in a certain sense independent of the choice of generators of H₁.

We propose a new method to enumerate alternating knots using a transfer matrix approach. We apply it to count numerically various objects, including prime alternating tangles with two connected components, up to order 18–22, and comment on the large-order behavior in connection with one of the authors's conjecture.

We refine a Le and Murakami uniqueness theorem for the Kontsevich Integral in order to specify the relationship between the two (possibly equal) main universal Vassiliev link invariants: the Kontsevich Integral and the perturbative expression of the Chern-Simons theory. As a corollary, we prove that the Altschuler and Freidel anomaly -that groups the Bott and Taubes anomalous terms- is a combination of diagrams with two univalent vertices and we explicitly define the isomorphism of which transforms the Kontsevich integral into the Poirier limit of the perturbative expression of the Chern-Simons theory for framed links, as a function of α

Let k: S¹ → S² be a generic immersion with n double points. We present an algorithm that assigns to k a partitioned n × n matrix over Z/2Z, and show that k gives rise to an orthogonal decomposition of (Z/2Z)ⁿ. We discuss a connection between this decomposition and the trip matrix of an alternating knot diagram produced from k.

We consider Dehn surgeries on some non-simple 2-component links in S³ which yield S³. As corollary we give a class of 2-component links which are determined by their complements.

Everyone knows that the Euler characteristic of a combinatorial manifold is given by the alternating sum of its numbers of simplices. It is shown that there are other linear combinations of the numbers of simplices which are combinatorial invariants, but that all such invariants are multiples of the Euler characteristic.

Let S be a connected open subset of 2-sphere S² which is identified with the extended plane R² ∪ {∞}. We assume that S contains the n segments {1, 2, …, n} × [-1, 1]. An n-lace ℓ (in S) is the union ℓ₁ ∪ … ∪ ℓ_n of disjoint simple arcs in S such that ∂ ℓ_i = {(i, 1), (π (i),-1)}, i = 1, …, n, for some permutation π of {1, 2, …, n}.

In this paper we will do present mapping class group description of n-laces, and 1-lace in 1-punctured plane, and 2-laces in S². We will also describe the isotropy subgroup of the trivial lace in .

We show that any tunnel number one knot group has a two generator one relator presentation in which the relator is a palindrome in the generators. We use this fact to compute the character variety for this knot groups and we show that it is an affine algebraic set .