## 2020年第七届中俄纽结会议

**The 7 ^{th} Chinese-Russian Conference on Knot Theory and Related Topics**

**Dec 3-6, 2020 @ ****Beijing Normal University**

**Program Committee**

Vassily Manturov (Moscow Institute of Physics and Technology)

Sergei Matveev (Chelyabinsk State University)

Andrei Vesnin (Tomsk State University, Sobolev Institute of Mathematics, Novosibirsk State University, Higher School of Economics)

Zhiyun Cheng (Beijing Normal University)

Jiajun Wang (Peking University)

Zhiqing Yang (Dalian University of Technology)

Ying Zhang (Soochow University)

**Organizing Committee**

Zhiyun Cheng, Hongzhu Gao (Beijing Normal University)

**Hosting Organizations**

Beijing Normal University

Dalian University of Technology

Moscow Institute of Physics and Technology

Peking University

Soochow University

Tomsk State University

**Dec 3, 2020 (The Main Building 1220****)**

13:55-14:00 Opening Speech

14:00-14:45 Yi Liu < Volume of Seifert representations for graph manifolds and finite covers >

14:45-15:00 Tea Break

15:00-15:45 Andrey Malyutin <Hyperbolic knots are not generic>

15:45-16:00 Tea Break

16:00-16:45 Ran Tao < Cosmetic surgery conjecture for some satellite knots>

16:45-17:00 Tea Break

17:00-17:45 Seongjeong Kim <3-free braids and integer valued index>

**Dec 4, 2020 (The Main Building 1220****)**

14:00-14:45 Nikolay Bogachev <Arithmetic and quasi-arithmetic hyperbolic reflection groups>

14:45-15:00 Tea Break

15:00-15:45 Haimiao Chen < SL(2,C)-character varieties of Montesinos knots>

15:45-16:00 Tea Break

16:00-16:45 Valeriy Bardakov < On residually finite quandles>

16:45-17:00 Tea Break

17:00-17:45 Xiao Wang <On Yang-Baxter homology>

**Dec 5, 2020 (The Main Building 1220****)**

14:00-14:45 Vassily Manturov < Braid actions, Legendrian links, domino tilings, Thompson groups and unsolved problems >

14:45-15:00 Tea Break

15:00-15:45 Xiaoming Du <The torsion generating set of the extended mapping class groups in low genus cases >

15:45-16:00 Tea Break

16:00-16:45 Igor Nikonov <Virtual index cocycles and invariants of virtual links>

16:45-17:00 Tea Break

17:00-17:45 Xiang Liu < Embeddings of templates in 3-spaces>

**Dec 6, 2020 (The Main Building 1220****)**

14:00-14:45 Qiang Zhang < Fixed point indices and fixed words at infinity of selfmaps of graphs >

14:45-15:00 Tea Break

15:00-15:45 Maxim Ivanov < F-polynomials and connected sum of virtual knots>

15:45-16:00 Tea Break

16:00-16:45 Jiming Ma < Three-manifolds at infinity of complex hyperbolic orbifolds >

16:45-17:00 Tea Break

17:00-17:45 Andrey Vesnin < Volumes of hyperbolic right-angled ideal polyhedra>

**Titles & Abstracts**

Valeriy Bardakov (Sobolev Institute of Mathematics, Novosibirsk State University, Tomsk State University)

**Title: **On residually finite quandles

**Abstract: **We introduce a residually finite quandles and show that free quandles and knot quandles are residually finite. Also, we prove that free products of residually finite quandles are residually finite provided their associated groups are residually finite. As associated groups of link quandles are link groups, which are known to be residually finite, it follows that link quandles are residually finite. This is joint work with Mahender Singh and Manpreet Singh.

Nikolay Bogachev (Skoltech & MIPT)

**Title: **Arithmetic and quasi-arithmetic hyperbolic reflection groups

**Abstract: **I will give a survey talk about arithmetic and quasi-arithmetic hyperbolic lattices (i.e. finite co-volume discrete groups of isometries in the hyperbolic space) generated by reflections. In particular, we will discuss the recent paper with A. Kolpakov (to appear in IMRN, arXiv: 2002.11445v2).

Haimiao Chen (Beijing Technology and Business University)

**Title:** SL(2,C)-character varieties of Montesinos knots

**Abstract:** We show how to determine the SL(2,C)-character variety for each Montesinos knot.

Xiaoming Du (South China University of Technology)

**Title: **The torsion generating set of the extended mapping class groups in low genus cases

**Abstract: **For the extended mapping class group of the closed orientable surfaces with genus 3 and 4, we prove that they can be generated by two elements of finite orders. For the case of the torus we prove it cannot. Together with the former result for the case of genus at least 5, we now get a rather neat solution for the problem.

Maxim Ivanov (Novosibirsk State University, Tomsk State University)

**Title:** F-polynomials and connected sum of virtual knots

**Abstract: **It is known that connected sum of two virtual knots is not uniquely determined and depends on knot diagrams and choosing the points to be connected. But different connected sums of the same virtual knots cannot be distinguished by Kauffman’s affine index polynomial.

In 2018 Kirandeep Kaur, Madeti Prabhakar and Andrei Vesnin introduced a generalization of affine index polynomial - a family of F-polynomials. For any pair of virtual knots K and M under some assumptions on knot M we construct an infinite family of different connected sums of K and which can be distinguished by F-polynomials.

Seongjeong Kim (Moscow institute of physics and technology, Moscow state technical university)

**Title:** 3-free braids and integer valued index

**Abstract: **In this talk we introduce a notion of 3-free braids and a mapping from pure braids to 3-free braids. We study study a $G_{n}^{3}$-like group corresponding to 3-free braids and an integer valued index will be constructed by using $G_{n}^{3}$-like group. In the end of talk, we discuss about geometrical meanings of 3-free braids and integer-valued indices.

Yi Liu (BICMR)

**Title: **Volume of Seifert representations for graph manifolds and finite covers

**Abstract: **Seifert volume is a topological invariant for closed orientable 3-manifolds. It is introduced by Brooks and Goldman as a generalization of the simplicial volume. In this talk, I will discuss some recent progress on this invariant for graph manifolds, and in particular, an effective formula that allows one to compute the volume of any representation into the motion group of the Seifert geometry. This is joint work with Pierre Derbez and Shicheng Wang.

Xiang Liu (Capital Normal University)

**Title: **Embeddings of templates in 3-spaces

**Abstract: **The template is topologically a compact branched surface, which is used to model knots and links as periodic orbits of a 3-dimensional flow. For embedded templates in S^3, we introduce isotopic invariants that can classify Smale flows on S^3 up to isotopy. We also consider the space of embeddings of a template with fixed homeomorphic type. It is joint work with XueZhi Zhao.

Jiming Ma (Fudan University)

**Title: **Three-manifolds at infinity of complex hyperbolic orbifolds

**Abstract:** We show the manifolds at infinity of the complex hyperbolic triangle groups $\Delta_{3,4,4;\infty}$ and $\Delta_{3,4,6;\infty}$, are one-cusped hyperbolic 3-manifolds $m038$ and $s090$ in the Snappy Census respectively. That is, these two manifolds admit spherical CR uniformizations.

Moreover, these two hyperbolic 3-manifolds above can be obtained by Dehn surgeries on the first cusp of the two-cusped hyperbolic 3-manifold $m295$ in the Snappy Census with slopes $2$ and $4$ respectively. In general, the main result in this paper allows us to conjecture that the manifold at infinity of the complex hyperbolic triangle group $\Delta_{3,4,n;\infty}$ is the one-cusped hyperbolic 3-manifold obtained by Dehn surgery on the first cusp of $m295$ with slope $n-2$. This is a joint work with Baohua Xie.

Andrey Malyutin (St.Petersburg Department of V.A.Steklov Institute of Mathematics of RAS)

**Title: **Hyperbolic knots are not generic

**Abstract:** A well-known conjecture in knot theory says that the proportion of hyperbolic knots among all of the prime knots of n or fewer crossings approaches 1 as n approaches infinity. We disprove this conjecture. This is joint work with Yury Belousov.

Vassily Manturov (Moscow Institute of Physics and Technology)

**Title: **Braid actions, Legendrian links, domino tilings, Thompson groups and unsolved problems

**Abstract**: In the present talk we aim to relate various objects in topology, algebra, and combinatorics.

We deal with braids which act on knots. The algebraic construction closely connected with this concept is the \Gamma_n^k groups which stem from Delaunay triangulations of surfaces (and similar partitionings of higher-dimensional spaces).

Along the lines of tiling a surface with triangles, one may consider the {\em domino tilings} — breaking the surfaces into $2 \times 1$ quadrilateral (or, more generally, into $k \times m$ quadrilaterals). Such tilings may undergo certain transformations (closely related to Ptolemy flips, Matveev-Piergalliny moves, Pachner moves and other well-known geometric transformations) and that naturally leads to the definition of new interesting groups generated by those moves.

We shall discuss the connections between the domino tilings theory with the Legendrian knots theory and with Thompson groups (which are known to be closely related to knots and their diagrams).

The talk shall also contain an overview of currently unsolved problems and research ideas in the field.

Igor Nikonov (Lomonosov Moscow State University)

**Title:** Virtual index cocycles and invariants of virtual links

**Abstract: **Virtual index cocycle is the 1-cochain that counts virtual crossings in the arcs of a virtual link diagram. We show how this cocycle can be used to reformulate and unify some known invariants of virtual links.

Ran Tao (Sichuan Normal University)

**Title:** Cosmetic surgery conjecture for some satellite knots

**Abstract: **Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. It is conjectured that nontrivial knots in S^{3} do not admit purely cosmetic surgeries. In this talk, we describe how to solve this problem for cable knots and composite knots.

Andrey Vesnin (Tomsk State University, Sobolev Institute of Mathematics, Novosibirsk State University, Higher School of Economics)

**Title:** Volumes of hyperbolic right-angled ideal polyhedra

**Abstract:** In the context of studding hyperbolic link complements we consider ideal right-angled polyhedral in a hyperbolic 3-space. We describe a census of those polyhedra by volumes and establish volume bounds. This is joint work with Andrey Egorov.

Xiao Wang (Jilin University)

**Title: **On Yang-Baxter homology

**Abstract: **As a generalization of rack homology, Yang-Baxter homology is promising to be useful for knot theory. In this talk, we will start by introducing the concepts of (bi)racks, (bi)quandles and their homology theories. Then we give the definition of the Yang-Baxter homology for column unital Yang-Baxter operators. Finally, we give some recent results, regarding to computations of the homology of a family of Yang-Baxter operators giving the HOMFLYPT polynomial.

Qiang Zhang (Xi'an Jiaotong University)

**Title:** Fixed point indices and fixed words at infinity of selfmaps of graphs

**Abstract: **Indices of fixed point classes play a central role in Nielsen fixed point theory. Jiang-Wang-Zhang proved that for selfmaps of graphs and surfaces, the index of any fixed point class has an upper bound called its characteristic.

In this talk, we study the difference between the index and the characteristic for selfmaps of graphs. First, for free groups, we extend attracting fixed words at infinity of automorphisms into that of injective endomorphisms. Then, by using relative train track technique, we show that the difference mentioned above is quite likely to be the number of equivalence classes of attracting fixed words of the endomorphism induced on the fundamental group. Since both of attracting fixed words and the existed characteristic are totally determined by endomorphisms themselves, we give a new algebraic approach to estimate indices of fixed point classes of graph selfmaps.

As consequence, we obtain an upper bound for attracting fixed words of injective endomorphisms of free groups, generalizing the one for automorphisms due to Gaboriau-Jaeger-Levitt-Lustig. Furthermore, we give a simple approach to roughly detecting whether fixed words exist or not.

This is joint work with ZHAO Xuezhi.

This conference is supported by** NSFC 11771042 **+** School of Mathematical Sciences, BNU**