Journal of Knot Theory and Its Ramifications

We show that there exist infinitely many 3-manifolds M, each of which admits a genus 4 Heegaard splitting V∪_PW such that there is a Scharlemann-Thompson untelescoping of V∪_PW which decomposes M into three pieces, and that any Casson-Gordon untelescoping of V∪_PW decomposes M into exactly two pieces.

Litherland has shown that if a knot is (+)-amphicheiral then its m-twist-spin is reversible. We show that, for classical knots, in many cases the converse holds.

We give a new simple proof for the weights of Ocneanu's trace on Iwahori–Hecke algebras of type A. This trace is used in the construction of the HOMFLYPT-polynomial of knots and links (which includes the famous Jones polynomial as a special case). Our main tool is Starkey's rule concerning the character tables of Iwahori–Hecke algebras of type A.

It is shown that two braids represent transversally isotopic links if and only if one can pass from one braid to another by conjugations in braid groups, positive Markov moves, and their inverses.

Let k be a non-negative integer. Then any embedding of the complete graph on 6·2^k vertices into a three-space contains a two-component link with the absolute value of its linking number greater than or equal to 2^k. Let j be a non-negative integer. Then any embedding of the complete graph on 48·2^j vertices into a three-space contains a knot with the absolute value of the second coefficient of its Conway polynomial greater than or equal to 2^2j.

An n-string tangle (B³,T) is a 3-ball B³ containing n properly embedded arcs T={t_i} and it is called rational if there is a homeomorphism of pairs from (B³,T) to (D,P)×I where D is the unit disk, P is any set of n points in the interior of D and I is the unit interval. In this article we partially generalize the well known classification of 2-string rational tangles to 3-string rational tangles. Using the results obtained we analyze knotted and linked products of site-specific recombination mediated by the Gin DNA invertase and give a possible solution to its action. Gin is an enzyme that carries out processive recombination.

We generalize some aspects of standard knot-theory to all ribbon-disc complements. We study asphericity of the complement of properly embedded links in certain contractible singular 3-manifolds that should be thought off as replacements of the 3-ball in the classical setting. We apply our results to show asphericity of 2-complexes modelled on labelled oriented graphs that correspond to alternating prime projections on some surface.

We provide an algorithm to decompose any diagram of a simply represented knot into an unlink diagram via certain local moves; such a decomposition exhibits an equivalent knot as a band-sum of represented unknots. This gives a represented ribbonlike surface respecting the knot's representation, extending the branch set to a smoothly embedded surface in the 4-ball.

Via application of the methods in [2], as extended in [3], we can thus solve the problem [1] of algorithmically obtaining a Kirby diagram for any 3-manifold presented as a finite-sheeted (not necessarily simple) covering of the 3-sphere branched over a knot (or graph).

A topological proper knot is a proper embedding f:ℝ¹→M³ of the real line into an open 3-manifold. Two proper knots are equivalent if they can be connected by a topological proper isotopy. In this paper, we answer a question posed by the author in [6] and show that, up to topological equivalence and orientation, all proper knots running between the opposite ends of D²×ℝ¹ are equivalent. Then sufficient conditions for a topological proper knot to be equivalent to a piecewise linear proper knot are given.

Properties of polynomial invariants Δ_i for oriented virtual links are established. The effects of taking mirror images and reversing orientation of the link diagram are described. The relationship between Δ₀(u,v) and an invariant of F. Jaeger, L. Kauffman, H. Saleur and J. Sawollek is discussed.

Sutured manifolds that are boundary-reducible and produce interval-bundles via 2-handle addition are characterized as product sutured manifolds.

In this paper, we show that for any torus giving a (1,1)-splitting of (M,K), every closed orientable surface of genus g>0 which is meridionally incompressible can be isotoped to intersect the the splitting torus in a circle. As an application, we estimate the distance, which is defined in [16], of (1,1)-splittings of (1,1)-knots whose exteriors contain closed orientable meridionally incompressible surfaces.