Journal of Knot Theory and Its Ramifications

Using basic topology and linear algebra, we define a plethora of invariants of boundary links whose values are power series with noncommuting variables. These turn out to be useful and elementary reformulations of an invariant originally defined by M. Farber [Fa2].

Let χ be an oriented link type in the oriented 3-sphere S³ or ℝ³ = S³ - {∞}. A representative X ∈ χ is said to be a closed braid if there is an unknotted curve A ⊂ S³ - X (the axis) and a choice of fibration ℋ of the open solid torus S³ - A by meridian discs {H_θ : θ ∈ [0, 2 π]}, such that whenever X meets a fiber H_θ the intersection is transverse.

Closed braid representations of χ are not unique, and Markov's well-known theorem asserts that any two are related by a finite sequence of elementary moves. The main result in this paper is to give a new proof of Markov's theorem. We hope that our new proof will be of interest because it gives new insight into the geometry.

We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed generically immersed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes of knot diagrams on surfaces. Using these bijections, we define concordance and link homology for virtual links. As an application, it is shown that Kauffman's example of a virtual knot diagram is not equivalent to a classical knot diagram.

We compute the slice euler characteristic of certain links by using the so-called generalized adjunction formula, which is proved by Kronheimer, Mrowka, Morgan, Szabó and Taubes. Furthermore, for links obtained from such links by band surgery, we estimate the unknotting numbers, the 4-dimensional clasp numbers, and the slice euler characteristic.

In this note, we will give an infinite family of simple fibered knots of genus 3 whose canonical genus an arbitrarily large.

In this note, we will study Δ link homotopy (or self Δ-equivalence), which is an equivalence relation of ordered and oriented link types. Previously, a necessary condition is given in the terms of Conway polynomials for two link types to be Δ link homotopic. A pair of numerical invariants δ₁ and δ₂ classifies all (ordered and oriented) prime 2-component link types with seven crossings or less up to Δ link homotopy. We will show here that for any pair of integers n₁ and n₂ there exists a 2-component link κ such that δ₁(κ) = n₁ and δ₂(κ) = n₂ provided that at least one of n₁ and n₂ is even.

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 (K_i, K_j) of the member of the family, the Gordian distance d_G(K_i, K_j) = 1.

We show that the linking signature of a closed oriented 4-manifold with infinite cyclic first homology is twice the Rochlin invariant of an exact leaf with a spin support if such a leaf exists. In particular, the linking signature of a surface-knot in the 4-sphere is twice the Rochlin invariant of an exact leaf of an associated closed spin 4-manifold with infinite cyclic first homology. As an application, we characterize a difference between the spin structures on a homology quaternion space in terms of closed oriented 4-manifolds with infinite cyclic first homology, so that we can obtain examples showing that some different punctured embeddings into S⁴ produce different Rochlin invariants for some homology quaternion spaces.

The HC-move was defined as an unknotting operation of a ribbon 2-knot. Representing a ribbon 2-knot by a virtual arc, we see that the HC-move corresponds to one of the "forbidden moves", which unknot every virtual knot. Using this, we consider the α₂-invariant of a ribbon 2-knot and a relation between the Δ-move for a 1-knot and the HC-move for its spun 2-knot.

Let K be a ribbon 2-knot. We show that for any ideal J of the Laurent polynomial ring Λ, if Alexander polynomail of K is trivial, i.e., Δ_K(t) = 1, then all colorings of K on Λ/J are trivial.

Given a tangle diagram of a classical knot K, we construct a surface diagram of any twist-spun 2-knot of K. From the crossings of the tangle diagram, we can get information of the corresponding triple points of the surface diagram, which are used to compute cocycle invariants of twist-spun 2-knots.

We construct minimal arc presentations of some non-alternating links which are adequate and generalise the class of semi-alternating links. The construction allows us to make some remarks on arc index of mutants.

This paper is an introduction to supercrossing index for knots and links, which is related to crossing index in the same way that N. Kuiper's superbridge index is related to bridge index. A variety of results on supercrossing index and the associated crossing map are given.

We show that the list {3₁, 4₁, 5₂, 6₁, 6₂, 6₃, 7₂, 7₃, 7₄, 8₄, 8₇, 8₉} contains all 3-superbridge knots. We also supply the best known estimates of the superbridge index for all prime knots up to nine crossings.