Journal of Knot Theory and Its Ramifications

This is a preface to the special issue on Proceedings of the Sixth Russian-Chinese Conference on Knot Theory and Related Topics which was held in Novosibirsk in June 2019.

In this paper, we study the singular pure braid group ${\mathrm{\text{SP}}}_n$ for $n=2,3$. We find generators, defining relations and the algebraical structure of these groups. In particular, we prove that ${\mathrm{\text{SP}}}_3$ is a semi-direct product ${\mathrm{\text{SP}}}_3={\tilde{V}}_3⋋\mathbb{Z}$, where ${\tilde{V}}_3$ is an HNN-extension with base group ${\mathbb{Z}}^2\ast {\mathbb{Z}}^2$ and cyclic associated subgroups. We prove that the center $Z({\mathrm{\text{SP}}}_3)$ of ${\mathrm{\text{SP}}}_3$ is a direct factor in ${\mathrm{\text{SP}}}_3$.

This paper is dedicated to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group ${\mathrm{\text{VP}}}_n$. We define simplicial group ${\mathrm{\text{VP}}}_{\ast }$ and its simplicial subgroup $T_{\ast }$ which is generated by ${\mathrm{\text{VP}}}_2$. Consequently, we describe ${\mathrm{\text{VP}}}_4$ as HNN-extension and find presentation of $T_2$ and $T_3$. As an application to classical braids, we find a new presentation of the Artin pure braid group $P_4$ in terms of the cabled generators.

In this paper, we discuss how to define a chord index via smoothing a real crossing point of a virtual knot diagram. Several polynomial invariants of virtual knots and links can be recovered from this general construction. We also explain how to extend this construction from virtual knots to flat virtual knots.

The knotting probability of an arc diagram is defined as the quadruplet of four kinds of finner knotting probabilities which are invariant under a reasonable deformation containing an isomorphism on an arc diagram. In a separated paper, it is shown that every oriented spatial arc admits four kinds of unique arc diagrams up to isomorphisms determined from the spatial arc and the projection, so that the knotting probability of a spatial arc is defined. The definition of the knotting probability of an arc diagram uses the fact that every arc diagram induces a unique chord diagram representing a ribbon 2-knot. Then the knotting probability of an arc diagram is set to measure how many nontrivial ribbon genus 2 surface-knots occur from the chord diagram induced from the arc diagram. The conditions for an arc diagram with the knotting probability 0 and for an arc diagram with the knotting probability 1 are given together with some other properties and some examples.

In the paper, we give an equivalent description of the lens space $L(p,q)$ with $p$ prime in terms of any corresponding Heegaard diagrams as follows: Let $M$ be a closed orientable 3-manifold with ${\pi }_1(M)\ne 1,$ and $U{\cup }_{F}V$ a Heegaard splitting of genus $n$ for $M$ with an associated Heegaard diagram $(U;J_1,\dots ,J_n)$. Assume $p$ is a prime integer. Then $M$ is homeomorphic to the lens space $L(p,q)$ if and only if there exists an embedding $f:U\hookrightarrow L(p,q)$ such that $L=\{f(J_1),\dots ,f(J_n)\}$ bounds a complete system of surfaces for $W=\overline{L(p,q)\backslash f(U)}$.

The twin group $T_n$ is a right angled Coxeter group generated by $n-1$ involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this paper, we study some properties of twin groups whose analogues are well known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups $T_n$ have $R_{\infty }$-property and are not co-Hopfian for $n\ge 3$.

The $d$-fold ($d\ge 3$) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions of these braid group representations into automorphism groups of free groups in terms of the actions on the generators of free groups. We also give a systematic way of constructing and expressing these braid group representations in terms of a new gadget, called covering groupoid. We prove that each generator ${\tilde{\beta }}_i$ of braid group inside mapping class group induced by $d$-fold covering is the product of $d-1$ Dehn twists on the surface.

Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in ${𝕊}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high-dimensional simplex in both the Gordian complexes, i.e. by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that the constructed virtual knots have the same affine index polynomial.

In this paper, we study the minimal symplectic elliptic surfaces $E(k)$ with homologically trivial symplectic finite group actions, and get a rigidity theorem under some restriction.

We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics.