## Second Russia-Korea Conference

on Knot theory and Related Topics

## List of speakers

3rd Nov. 2020

Andrei Vesnin, Tomsk State University

Title:

Abstract:

Jieon Kim, Pusan National University

Title: Unoriented tribracket counting invariants of surface-links

Abstract: A surface-link is a surface smoothly embedded in 4-space. When every component of a surface-link is oriented, it is called an oriented surface-link. Otherwise, it is called an unoriented surface-link.

In 2018, myself and S. Nelson defined invariants for oriented surface-links by using a biquasile, which

is used to define invariants of oriented classical links via their dual graph diagrams. Biquasiles can be understood as a special case of the ternary algebraic structures. Here we call it a vertical tribracket.

In this talk, we introduce a special case of vertical tribrackets, called an unoriented tribracket. By using unoriented tribrackets, we define an invariant of unoriented surface-links.This is a joint work with S. Y. Lee and S. Nelson.

Seung Yeop Yang, Kyungpook National University

Title: Normalized set-theoretic Yang-Baxter homology of a biquandle

Abstract: Biracks and biquandles, which are useful for studying low-dimensional topology, especially knot theory, are special families of solutions to the set-theoretic Yang-Baxter equation. A homology theory for the set-theoretic Yang-Baxter equation was introduced by Carter, Elhamdadi, and Saito. In this talk, we compute the normalized set-theoretic Yang-Baxter homology groups of cyclic biquandles and Alexander biquandles and study their applications.

Mohd Ibrahim, Pusan National University

Title:

Abstract:

Kodai Wada, Osaka University

Title: Classification of 2-component virtual link up to Xi-moves

Abstract: The Xi-move is a local move which refines the usual forbidden moves in virtual knot theory. This move was introduced by Satoh and Taniguchi, who showed that it characterizes the information contained by the odd writhe of virtual knots, a fundamental invariant defined by Kauffman. In this talk, we extend this result by classifying 2-component virtual links up to Xi-moves, using refinements of the odd writhe and linking numbers. This is a joint work with Jean-Baptiste Meilhan and Shin Satoh.

Jinseok Oh, Kyunpook National University

Title: On torsion in homology of a dihedral quandle.

Abstract: Many studies have investigated rack and quandle homology computations. The free parts of the homology were completely determined, but little is known about the corresponding torsion parts, although some details regarding the orders of torsion elements in rack homology have been derived. For example, it is known that the order of a finite quandle annihilates its reduced quandle homology and the torsion subgroup of its rack homology if the quandle is a quasigroup. In this talk, we investigate a relationship between the torsion subgroup of homology of a dihedral quandle and its automorphism group. This is joint work with Seung Yeop Yang.

Minju Seo, Pusan National University

Title: Quandle coloring quivers of surface-links

Abstract: here

4th Nov. 2020

Sergei Gukov, California Institute of Technology

Title:

Abstract:

Daniil Rudenko, University of Chicago

Title:

Abstract:

Kanako Oshiro, Sophia University

Title: Goeritz matrices and Dehn colorings of spatial graphs

Abstract: In this talk, we will introduce the Goeritz matrix of a spatial graph whose vertices are of even degree. We will show a relationship between Goeritz matrices and Dehn colorings of spatial graphs. This study is a joint work with Natsumi Oyamaguchi(Shumei University).

Svetlana Bezgodova and Denis P. Ilyutko, Lomonosov Moscow State University

Title: Emdeddings of 4-graphs with cross strucure into 2-surfaces and minimal forbidden minors

Abstract:

Naoko Kamada, Nagoya City University

Title: Pseudo Goeritz martices for virtual link

Abstract: A Goeritz martix is defined for a classical link by suing a checkerboard coloring of its link diagram.

For a virtual link presented by a virtual link diagram admitting a checkerboard coloring,

Young Ho Im, Kyeounghui Lee, and Sang Youl Lee introduces a modified Goeritz matrix.

However, not every virtual link has a diagram which admits a checkerboard coloring.

In this talk, we introduce the notion of a pseudo Goeritz matirx for any virtual link diagram and we obtain a virtual link invariant derived from the matrix.

Sanghoon Park, Pusan National University

Title: Link Invariants from Quandle Coloring Quivers Using Combinatorial Laplacians

Abstract: here

Suhyeon Jeong, Pusan National University

Title: Psybrackets, Pseudoknots and Singular knots

Abstract: here

5th Nov. 2020

Vassily O. Manturov* and Denis A. Fedoseev**

Moscow Institute of Physics and Technology*, Moscow State University**

Title: On 3-free links and link-homotopy invariants

Abstract:

About ten years ago, the first named author introduced the theory of free knots (previously conjectured by Turaev to be trivial) and found that this theory --- a very rough simplification of the theory of virtual links --- admitted new types of invariants never seen before: the invariants of links are valued in {\em pictures}, more precisely, in linear combinations of knot diagrams.

For some classes of links, we have the formula $$[K]=K,$$ where $K$ in the LHS is our favourite link diagram (which is subject to various Reidemeister-like moves), and $K$ in the RHS is the same diagram but seen as the rigid object.

After some time it was noted that this approach does not work immediately for classical knots. In fact, the reason is that the approach when we look at ``nodes'' being ``double'' classical crossings is not the best one for classical knots. It is much better to look at ``triple'' crossings.

In the present talk, we construct a map from equivalence classes of closed braids to 3-free knots and links (elder brothers of free knots and links). We also consider the map from {\em links up to link-homotopy} to 3-free links modulo some moves.

This talk is based on a joint work with S. Kim.

Partially supported by RFBR grants 20-51-53022 and 19-51-51004.

Sera Kim, Pusan National University

Title: On the polynomial invariants and its applications

Abstract: here

Seokbeom Yoon, Korea Institute for Advanced Study

Title:

Abstract:

Nikolay Abrosimov, Sobolev Institute of Mathematics

Title: On volumes of hyperbolic cone manifolds 52(α) and 732(α,β)

Abstract:

We obtain exact formula for the volume of hyperbolic cone manifolds 52(α). First, we derive the relation between the complex length of the singular geodesic and conical angle α in the form of Cotangent rule. Then by making use of A-polynomial of the knot 52 and Schlafli differential equation we prove the volume formula.

Also, we present explicit formula for the volume of hyperbolic cone manifolds 732(α,β). The proof of the latter formula is based on the background given in the paper of D. Derevnin, A. Mednykh and M. Mulazzani (2004).

This is a joint work with Alexander Mednykh.

Yuta Taniguchi, Osaka City University

Title: Quandle coloring quivers using dihedral quandles

Abstract: The quandle coloring quiver which was introduced by K. Cho and S. Nelson is a directed graph-valued invariant for oriented links. This invariant can distinguish links which have the same quandle coloring number. In this talk, we will show that, when we use a dihedral quandle, the quandle coloring quivers are determined by the algebraic structure of the set of quandle colorings. As a corollary, we will show that, when we use a dihedral quandle of prime order, the quandle coloring quivers are equivalent to the quandle coloring numbers.

6th Nov. 2020

Seiichi Kamada, Osaka University

Title: On chart descriptions of branched coverings, surface foldings and braided surfaces

Abstract: In this talk we discuss methods of describing branched coverings, surface foldings and braided surface, called chart descriptions, in order to clarify relationship among those topological objects. A branched covering of a surface is simply determined by its monodromy representation to the symmetric group. A (permutation) chart of a branched covering is a labelled graph on the base space $\Sigma$ which induces the monodromy representation. Once a permutation chart is given, we may construct a surface in $I \times \Sigma$ in a special form called a surface folding which is a lift of the branched covering. Furthermore, when a permutation chart is lift to a braid chart, the surface folding is lift to braided surface in $I^2 \times \Sigma$. Chart description plays a fundamental role in a research project with J. S. Carter, R. Piergallini and D. Zuddas.

Neha Nanda, Indian Institute of Science Education and Research Mohali

Title: Virtual twins and doodles

Abstract: here

Seonhwa Kim, KIAS

Title: Diagrammatic formulas for geometric knot invariants from the volume conjecture.

Abstract: Geometric knot invariants like hyperbolic volume are defined independent of a knot diagram. There have been excellent algorithms like SnapPEA computing hyperbolic volume whose input is a knot diagram. However, the relationship between a knot diagram and hyperbolic volume is still mysterious and uncovered. On the other hand, quantum knot invariants like Jones polynomial are usually defined through a knot diagram, unlike the definition of geometric invariants. The famous volume conjecture in quantum topology is about the relation between colored Jones polynomial and hyperbolic volume. Although the conjecture itself turned out to be very difficult to prove rigorously, this question gave us other viewpoints and fruitful results for the study on geometric invariants. From this perspective, I will introduce several concrete and elegant formulas for geometric invariants like hyperbolic volume, which would not have been found without the volume conjecture. Interestingly, those formulas, although for geometric invariants, have a form reminiscent of the diagrammatic state-sum of quantum invariants. If the time permits, I will also talk about several further questions in this line.

Igor M. Nikonov, Lomonosov Moscow State University

Title:

Abstract:

Olga D. Frolkina, M.V.Lomonosov Moscow State University

Title: Cantor sets with high-dimensional projections: existence and exceptionality

Abstract: here

Scott Carter

Title: Braiding Manifolds in codimension 2.

Abstract: The talk will consist of several examples of braidings of 3 and 4 dimensional manifolds in 5 and 6 space respectively.

The atomic pieces for such braidings will be discussed in a higher categorical context. This is based upon joint work with Seiichi Kamada.