Topology Project
Fundamental Groups, Representations and Geometric Structures in 3-Manifold Topology
November 21-23, 2016
Room E002, Building-E of Faculty of Science, Hiroshima University
Organizers: Teruaki Kitano, Yuya Koda and Takayuki Morifuji.
This workshop is supported by JSPS KAKENHI Grant Numbers 16H02145, 16K05161, 26800028.
Schedule
Monday, November 21
9:50–10:00 Opening
10:00–11:00 Yi Liu (Peking University/BICMR)
Geometric pieces and virtual representation volume (1) 11:20–12:20 Makoto Sakuma (Hiroshima University)
Kleinian groups generated by two parabolic transformations 13:50–14:20 Mizuki Fukuda (Tohoku University)
Distinguishing branched twist spins by knot determinants 14:30–15:00 Naoki Sakata (Hiroshima University)
Veering triangulations of hyperbolic fibered two-bridge link com- plements
15:20–16:20 Fumikazu Nagasato (Meijo University)
Ghost characters, character varieties and abelian knot contact homology
16:40–17:40 Takayuki Morifuji (Keio University)
Twisted Alexander polynomials of hyperbolic knots and links (1)
Tuesday, November 22
10:00–11:00 Shicheng Wang (Peking University) Mapping degrees and volume of presentations (1) 11:20–12:20 Masaaki Suzuki (Meiji University)
Epimorphisms between two-bridge knot groups and their crossing numbers
13:50–14:20 Kazuhiro Ichihara (Nihon University) The SL(2,C) Casson invariant and cosmetic surgeries 14:30–15:00 Evgeny Fominykh (Chelyabinsk State University)
Turaev-Viro invariants and minimal triangulations of 3-manifolds 15:20–16:20 Yi Liu (Peking University/BICMR)
Geometric pieces and virtual representation volume (2) 16:40–17:40 Takayuki Morifuji (Keio University)
Twisted Alexander polynomials of hyperbolic knots and links (2)
Wednesday, November 23
10:00–10:30 Ken’ichi Yoshida (University of Tokyo)
Union of 3-punctured spheres in a hyperbolic 3-manifold 10:40–11:10 Andrei Vesnin (Sobolev Institute of Mathematics)
Small covers of right-angled hyperbolic 3-orbifolds 11:30–12:30 Shicheng Wang (Peking University)
Mapping degrees and volume of presentations (2)
Abstracts
Evgeny Fominykh (Chelyabinsk State University): Turaev- Viro invariants and minimal triangulations of 3-manifolds
Abstract:
In this talk we construct minimal truncated triangulations for an infinite family of hyperbolic 3-manifolds with totally geodesic bound- ary. The proof of minimality is based on calculating of Turaev-Viro invariants.
Mizuki Fukuda (Tohoku University): Distinguishing branched twist spins by knot determinants
Abstract:
A branched twist spin is a 2-knot which appeared in the study of locally smooth circle actions on the 4-sphere. Pao and Plotnick showed that a 2-knot is a branched twist spin if and only if it is a fibered 2- knot which has a periodic monodromy. In this talk, we give a sufficient condition to distinguish non-trivial branched twist spins by using the first elementary ideals of them.
Kazuhiro Ichihara (Nihon University): The SL(2,C) Casson in- variant and cosmetic surgeries
Abstract:
I will talk about the SL(2,C) Casson invariant for 3-manifolds, and its applications to the cosmetic surgery problem for knots in the 3- sphere. In particular, in terms of boundary slopes, a condition for knots to admit no cosmetic surgeries will be given. This talk is based on a joint work with Toshio Saito (Joetsu University of Education).
Yi Liu (Peking University/BICMR):Geometric pieces and vir- tual representation volume
Abstract:
When an oriented 3-manifold contains a geometric piece of hyper- bolic or Seifert geometry, the manifold has a finite cover with positive representation volume with respect to that geometry. This result has
been shown in the joint work of the speaker with Pierre Derbez and Shicheng Wang. In these two talk, we explain the idea of the proof and discuss some details of the construction.
Takayuki Morifuji (Keio University): Twisted Alexander poly- nomials of hyperbolic knots and links
Abstract:
In the first talk we review the definition and basic properties of the twisted Alexander polynomial associated to an SL(2,C)-representation of a knot group. In the second talk we explain a conjecture of Dunfield, Friedl and Jackson for a hyperbolic knot, and then generalize it to a hyperbolic link in the 3-sphere. The talks are partially based on joint works with Taehee Kim and Takahiro Kitayama, and also with Anh T. Tran.
Fumikazu Nagasato (Meijo University): Ghost characters, char- acter varieties and abelian knot contact homology
Abstract:
We introduce a ghost character of a knot K using the trace-free charactersS0(K) of SL(2,C)-representations of the knot groupG(K).
This gives a tool to describe exactly the relationship of degree 0 abelian knot contact homologyHC0ab(K) with the character varietyX(Σ2K) of the 2-fold branched cover Σ2Kof 3-sphere branched alongK. Using this, we give some criteria to check
1. whether Ng’s conjecture concerned with the above relationship holds true or not,
2. when the map Φ :b S0(K) → X(Σ2K) constructed in [F.
Nagasato-Y. Yamaguchi: On the geometry of the slice of trace- free SL2(C)-characters of a knot group, Math. Ann. 354(2012), 967–1002] is not surjective.
Naoki Sakata (Hiroshima University): Veering triangulations of hyperbolic fibered two-bridge link complements
Abstract:
Agol proved that each punctured surface bundle over the circle with pseudo-Anosov monodromy, such that all complementary regions of the stable lamination have a puncture, admits a unique “veering” and layered triangulation. In this talk, I will explain a condition for the Epstein-Penner decomposition of each hyperbolic fibered two-bridge link complement to be veering with respect to the fiber structure.
I also hope to discuss veering triangulations of punctured surface bundles with pseudo-Anosov monodromy obtained from Penner’s con- struction.
Makoto Sakuma (Hiroshima University): Kleinian groups gen- erated by two parabolic transformations
Abstract:
At a workshop held in Budapest in 2002, Ian Agol announced 1. a classification of non-free Kleinian groups generated by two
parabolic transformations, and
2. a classifications of parabolic generating pairs of each of such Kleinian groups.
I would like to give a progressive report on my project to give a com- plete proof to the announcement. Part of this topic is a joint work with Shunsuke Aimi and Donghi Lee. I would also like to talk about a conjectural picture of the space of Kleinian groups generated by two parabolic transformations. This part is a joint work in progress with Gaven Martin.
Masaaki Suzuki (Meiji University): Epimorphisms between two- bridge knot groups and their crossing numbers
Abstract:
In this talk, we study the relationship between epimorphisms of two-bridge knot groups and their crossing numbers. In particular, if there exists an epimorphism from the knot group of a two-bridge knot K onto that of another knot K0, then the crossing number of K is
greater than or equal to three times of that of K0. By using it, we estimate how many knot groups a two-bridge knot group maps onto.
Moreover, we formulate the generating function which determines the number of two-bridge knot groups admitting epimorphisms onto the knot group of a given two-bridge knot.
Andrei Vesnin (Sobolev Institute of Mathematics): Small cov- ers of right-angled hyperbolic 3-orbifolds
Abstract:
Right-angled polyhedra in hyperbolic spaces are serving as useful building blocks for constructing hyperbolic manifolds and orbifolds with interesting properties. We will look at dimension three, where we describe existence conditions and the volume set structure. Then we will present a way to construct closed hyperbolic 3-manifolds re- lated to color-epimorphisms of right-angled Coxeter groups and find 2-fold branched coverings of the 3-sphere. As a particular case, we will discuss the first example of a closed orientable hyperbolic 3-manifold constructed by F. Loebell in 1931.
Shicheng Wang (Peking University): Mapping degrees and vol- ume of presentations
Abstract:
I will give two talks around the title. The first one is a survey, mainly on the recent results joint with Derbez, Liu and Sun. Profes- sor Liu Yi might talk on the construction of virtually positive volume representations. So my second talk might on some other results men- tioned in my survey.
Ken’ichi Yoshida (University of Tokyo): Union of 3-punctured spheres in a hyperbolic 3-manifold
Abstract:
An essential 3-punctured sphere in a hyperbolic 3-manifold is iso- topic to a totally geodesic one. We will classify the topological types for components of union of the totally geodesic 3-punctured spheres in an orientable hyperbolic 3-manifold. There are special types each of which appears in precisely one manifold.
