Journal of Knot Theory and Its Ramifications

The purpose of this note is to show that classical cobordism arguments, which go back to the pioneering works of Mandelbaum and Moishezon, provide quick and unified proofs of any knot surgered compact simply-connected 4-manifold $X_K$ becoming diffeomorphic to $X$ after a single stabilization by connected summing with $S^2\times S^2$ or $ℂ\mathbb{P}{}^2\#\overline{ℂ\mathbb{P}}{}^2$, and almost complete decomposability of $X_K$ for many almost completely decomposable $X$, such as the elliptic surfaces.

In this work, we study the decomposability property of branched coverings of degree $d$ odd, over the projective plane, where the covering surface has Euler characteristic $\le 0$. The latter condition is equivalent to say that the defect of the covering is greater than $d$. We show that, given a datum $\mathcal{𝒟}=\{D_1,\dots ,D_s\}$ with an even defect greater than $d$, it is realizable by an indecomposable branched covering over the projective plane. The case when $d$ is even is known.

A branched covering surface-knot over an oriented surface-knot $F$ is a surface-knot in the form of a branched covering over $F$. A branched covering surface-knot over $F$ is presented by a graph called a chart on a surface diagram of $F$. For a branched covering surface-knot, an addition of 1-handles equipped with chart loops is a simplifying operation which deforms the chart to the form of the union of free edges and 1-handles with chart loops. We investigate properties of such simplifications.

We consider involutory virtual biracks with good involutions, also known as symmetric involutory virtual biracks. Any good involution on an involutory virtual birack defines an enhancement of the counting invariant. We provide examples demonstrating that the enhancement is stronger than the unenhanced counting invariant.

We present a generalization of the Fox formula for twisted Alexander invariants associated to representations of knot groups over rings of $S$-integers of $F$, where $S$ is a finite set of finite primes of a number field $F$. As an application, we give the asymptotic growth of twisted homology groups.

Every surface bundle with genus $g$ fiber has a canonical Heegaard splitting of genus $2g+1$. In this paper, we discuss the topological indices of such Heegaard surfaces and prove the canonical Heegaard splitting of a surface bundle is topologically minimal if and only if it is critical, that is, its topological index is 2.

In a remark in his seminal 1987 paper, Jones describes a way to define the Burau matrix of a positive braid using a metaphor of bowling a ball down a bowling alley with braided lanes. We extend this definition to allow multiple bowling balls to be bowled simultaneously. We obtain representations of the Iwahori–Hecke algebra and a cabled version of the Temperley–Lieb representation.

A cord is a simple curve on a punctured plane. We introduce diagrams which represent isotopy classes of cords. Using such diagrams, we prove that there is a one-to-one correspondence from the set of the isotopy classes of cords to a set of symmetric matrices whose components are non-negative integers, and we give necessary conditions for matrices to represent the isotopy classes of cords.