International Workshop on Low-dimensional Topology

Date: May 6-7, 2016

Place: DUT International Conference Center, Dalian, China

Host: School of Mathematical Sciences, Dalian University of Technology

Co-Host: School of Mathematics, Liaoning Normal University

Invited Speakers:

Prof. Yukio Matsumoto, Gakushuin University, Japan

Prof. Akio Kawauchi, Osaka City University, Japan

Prof. Mikio Furuta, University of Tokyo, Japan

Prof. Seiichi Kamada, Osaka City University, Japan

Prof. Kazuo Habiro, Kyoto University, Japan

Prof. Ki Hyoung Ko, KAIST, Korea

Prof. Gyo Taek Jin, KAIST, Korea

Prof. Yongjin Song, Inha Univeristy, Korea

Prof. Andrei Yurievich Vesnin, Sobolev Institute of Mathematics, Russia

Prof. Velariy Bardakov, Novosibirsk State University, Russia

Prof. Fuquan Fang, Capital Normal University, China

Prof. Xuezhi Zhao, Capital Normal University, China

Prof. Ruifeng Qiu, East China Normal University, China

Prof. Zhiqing Yang, Dalian University of Technology, China

Prof. Xianan Jin, Xiamen University, China

Organizing Committee:

Jie Wu, National University of Singapore

Ximin Liu, Dalian University of Technology

Fengchun Lei, Dalian University of Technology

Contact: Ms. Fengling Li, dutlfl@163.com

Homepage: http://math.

Hotel: DUT International Conference Center (DUTICC) (neighbor to the South Gate of DUT),  http://hotel.

Conference Venue: Conference Room 2, DUT International Conference Center

Transportation: Taking a taxi is the easiest way from Dalian International Airport, or Dalian Train Station, or Dalian North Train Station to the hotel, and the taxi fare is about 30, 35, or 45 Chinese Yuan, respectively.

Sponsors: The workshop are supported in part by Dalian University of Technology, and grants (No.11329101 and No. 41171151) from NSFC.

Timetable

Category of handlebody embeddings

Titles and Abstracts

Speaker: Velariy Bardakov (Novosibirsk State University, Russia)

Title: Some representations of virtual braid group by automorphisms

Abstract: Braid group B_n has a faithful representation in the automorphism group Aut(F_n) of free group F_n of rank n (Artin's representation). Using this representation, one cans define a group of link that is a strong invariant of a link. Virtual braid group VB_n is a generalization of braid group. This group is the foundation of the Virtual knot theory. We describe some known representations of VB_n into Aut(G_n) for some group G_n. After this we introduce a new representation which is generalized the previous representation. Using this representation we introduce the group of virtual link and prove that this group is an invariant of the link.

Speaker: Fuquan Fang (Capital Normal University, China)

Title: Reflections in Riemannian manifolds

Abstract:

Speaker: Mikio Furuta (University of Tokyo, Japan)

Title: Seiberg-Witten theory, generalized (co)homology and TFT

Abstract: Formulatons of Floer homotopy type for Seiberg-Witten theory are given by Manolescu, Manolescu-Kronheimer and Khandhawit-Lin-Sasahira. We will discuss two related topics.:(1) formulations of TFT for Seiberg-Witten theory using generalized (co)homology theories. (2) an approach to Seiberg-WItten FLoer homotopy type when an obstruction vanishes. (Joint works with Tian Jun Li (1), and  Khandhawit and Sasahira (2)).

Speaker: Kazuo Habiro (Kyoto University, Japan)

Title: Category of handlebody embeddings

Abstract: I plan to talk about the category $H$ of handlebody embeddings. It has as objects nonnegative integers, and as morphisms from $m$ to $n$ embeddings of a handlebody of genus $m$ into a handlebody of genus $n$ modulo isotopy. Here embeddings and isotopies preserves "base discs" which are embedded disks in the boundaries of handlebodies. I will describe the braided monoidal category structure of $H$ and functors defined on $H$, which are regarded as invariants of handlebody embeddings.

Speaker: Gyo Taek Jin (KAIST, Korea)

Title: Examples and Counterexamples of the quadrisecant approximation conjecture

Abstract: We show smooth knots and polygonal knots, trivial and nontrivial, on which the quadrisecant approximation conjecture holds. We also show the counterexamples created by Bai-Wang-Wang, and discuss a possible modificaton of the conjecture.

Speaker: Seiichi Kamada (Osaka City University, Japan)

Title: Classification of 1-handles attaching to surface-links using quandles

Abstract: Classification of 1-handles attaching to surface-knots using groups was done by J. Boyle for oriented surface-knots, and by myself for non-orientable surface-knots. Here we discuss classification of 1-handles using quandles. We introduce the notion of the tensor product of quandles. Then 1-handles are naturally understood via the tensor product of the knot quandles, or the knot symmetric quandles.

Speaker: Akio Kawauchi (Osaka City University, Japan)

Title: On a cross-section of an immersed sphere-link in 4-space

Abstract: The torsion Alexander polynomial, the reduced torsion Alexander polynomial and the local signature invariant of a cross-section of an immersed sphere-link are investigated from the viewpoint of how to influence to the immersed sphere-link. It is shown that the torsion Alexander polynomial of a symmetric middle cross-section of a ribbon sphere-link is an invariant of the ribbon sphere-link. A generalization to a symmetric middle cross-section of an immersed ribbon sphere-link is given.

Speaker: Ki Hyoung Ko (KAIST, Korea)

Title: Automorphism groups of a family of non-rigid Artin groups

Abstract: An Artin group is rigid if its defining graph is unique. There have been extensive researches on automorphisms on rigid Artin groups such as Artin groups of finite types, free groups, and right-angled Artin groups. We will discuss automorphisms of a family of non-rigid Artin groups studied by John Crisp. In fact, we will completely determine the structures of automorphism groups of this family and this is one of the first results in this direction.

Speaker: Yukio Matsumoto (Gakushuin University, Japan)

Title: Riemann surfaces and crystallographic groups

Abstract: A crystallographic group is an isometry group acting on a Euclidean space $\mathbb{E}^n$ whose translation subgroup forms an $n$-dimensional lattice. For example, a so-called wall paper group is a two dimensional crystallographic group. This talk will report our recent discovery that certain crystallographic groups on $\mathbb{E}^{3g-3}$ naturally arize from  Teichm\"uller space of Riemann surfaces of genus $g$.

Speaker: Ruifeng Qiu (East China Normal University, China)

Title: On distance of Heegaard splitting

Abstract: In this talk, I will talk about some results on distance of Heegaard splitting, then try to explain why the definition "distance" led some important progresses on Heegaard splitting.

Speaker: Yongjin Song (Inha University, Korea)

Title: Embedding problems of Artin groups of type E

Abstract: It was proved by Wajnryb that there is no geometric embedding of Artin groups of type E into mapping class group of surface. In this talk we will investigate the possibility of existence of nongeometric embedding of Artin groups of type E.  

Speaker: Andrei Vesnin (Sobolev Institute of Mathematics, Novosibirsk, Russia)

Title: On complexity and Turaev-Viro invariants of 3-manifolds

Abstract: We will discuss new results on Matveev’s complexity of infinite families of orientable hyperbolic 3-manifolds. We will demonstrate how Turaev–Viro invariants of hyperbolic 3-manifolds with totally geodesic boundary can be used to find complexity of manifolds.

Speaker: Zhiqing Yang (Dalian University of Technology, China)

Title: An infinite-variable knot invariant

Abstract: This is a follow-up work of arXiv:1004.2085. The author modifed earlier work to get a stronger invariant. It uses a system of skein equations to define. It is a generalization of HOMFLY and Kauffman two variable polynomials. One simplified version of it is an infinite-variable HOMFLY polynomial.

Speaker: Xuezhi Zhao (Capital Normal University, China)

Titlt：On classification of cyclic orientation-reversing actions of big order on closed surfaces

Abstract: At the end of 19 century, A. Wiman proved that the order of any orientation-preserving periodic self-homeomorphism of a closed orientable surface of genus $g >1$ does not exceed $4g+2$. Later in the 1960s, W. Harvey showed that this maximum possible order is attained for each $g$. In the middle of the 1980s, J. J. Etayo showed that any finite cyclic group $Z_N$ generated by an orienta\-tion-re\-ver\-sing periodic self-homeomorphisms of a closed orientable surface $S_g$ of genus $g>1$ has order bounded above by $4g+4$ and $4g-4$ for $g$ even and odd respectively, and these bounds are sharp for all $g$. Five years later, S. Wang proved these results more directly in a purely topological way. The question to which extent the constructions of Etayo and Wang are unique was the original motivation for the present paper. Here we classify up to topological conjugation orientation-reversing actions of a cyclic group $Z_N$ on $S_g$, in function of a possible type of the quotient orbifold $S_g/\Z_N$, provided that $N>2g-2$.  In particular, we prove that Etayo-Wang extremal actions are unique up to topological conjugations.

Speaker: Xiao Guo (Harbin Institute of Technology, China)

Title: Parabolic polynomials of 2-bridge knots

Abstract: In the talk, I will introduce an algorithm for calculating the $p$-polynomials of $2$-bridge knots.

Speaker: Xianan Jin (Xiamen University, China)

Title: A relation between the Tutte polynomial and the HOMFLY polynomial with applications in DNA polyhedra

Abstract: Let $G$ be a plane graph. Let $D(G)$ be the oriented link obtained from $G$ by replacing each edge $e$ of $G$ by an alternatingly oriented 2-tangle $T_e$. We first establish a relation between the HOMFLY polynomial of $D(G)$ and the edge-weighted Tutte polynomial of $G$ by assigning suitable edge weights which depend on $T_e$'s or equivalently, the chain polynomial of $G$ with labels on edges of $G$. This relation extends works of F. Jaeger and L. Traldi. Then we apply the relation to DNA polyhedral links, the mathematical model for DNA polyhedra synthesized by chemists and biologists. To deal with complicated double crossover DNA 3-regular polyhedral links, another relation is further established for computing the chain polynomial of the truncated cubic graph with two different labels from the chain polynomial of the original labeled cubic graph by substitutions.

Speaker: Liang Liang (Liaoning Normal University, China)

Title: A sufficient condition for distance degenerating handle addtions to be bounded

Abstract: Let $M = V\bigcup_{s} W$ be a Heegaard splitting of 3-manifold $M$ and $F$ a component of $\partial M$ lying in $\partial_{-}V$. A simple closed curve $J$ in $F$ is called to be distance degenerating if the distance of $M = V\bigcup_{s} W$ is less than the distance of $M_{J} = V_{J}\bigcup_{s} W$. In this report, I will introduce the handle additions and Heegaard splittings of 3-manifolds. At last, I

will give a result about distance degenerating handle additions. This is a joint work with Fengchun Lei and Fengling Li.

Speaker: Naoko Kamada (Nagoya City University, Japan)

Title: Converting virtual knot diagrams to normal diagrams

Abstract: A virtual knot diagram is said to be normal when the corresponding abstract knot diagram is checkerboard colorable. Not every virtual knot diagram is normal, while all classical knot diagrams are normal. In this talk we discuss a method of converting a virtual knot diagram to a normal virtual diagram. We discuss an invariant of a virtual knot that can be obtained from an invariant of the normal virtual diagram.

Speaker: Zhiguo Li (Dalian University of Technology, China)

Title: On the unknotting number of welded knots

Abstract: We discuss the unknotting number of welded knots, and give a upper-bound of the unknotting number of welded knots by using the warping degree method, and  a lower bound of the unkotting number of welded knots by quandle colorings.

Speaker: Jiming Ma (Fudan University, China)

Title: Hyperbolicity of a random link via bridge position

Abstract:We show that a random link via random bridge position is hyperbolic, this is a joint work with Kazuhiro Ichihara.

Speaker: Wenyuan Yang (Peking University, China)

Title: Purely exponential growth of cusp-uniform actions

Abstract: In this talk, I will discuss the purely exponential type of the orbit growth function with connections to the finiteness of Bowen-Margulis-Sullivan measures. Our setup is to consider a cusp-uniform action of a countable group on a $\delta$-hyperbolic space. The main result is characterizing the purely exponential growth type of growth by a condition introduced by Dal'bo-Otal-Peign\'e. This condition is equivalent to the finiteness of Bowen-Margulis-Sullivan measures on the unit tangent bundle of geometrically finite Cartan-Hadamard manifolds with pinched negative curvature. In this case, our result recovers a theorem of Roblin (in a coarse form).

Speaker: Yanqing Zou (Dalian Nationalities University, China)

Title: The subset of $R^{3}$ realizing metrics on the curve complex

Abstract: It is known that the curve complex with the metric defined by Minsky and Masur is $\delta$-hyperbolic. Now we consider the possible metrics on the curve complex, which are built from a subset $V\subset R^{3}$, and discuss some properties of the curve complex under those new metrics. This is a joint work with Ruifeng Qiu and Faze Zhang.

School of Mathematical Sciences

Dalian University of Technology

                                                   Updated by April 25, 2016