## セミナー -- Low-dimensional Topology Seminar

Date

2015年12月16日(水) 14:30 -- 15:30

Room

京都大学 数理解析研究所 110号室

Speaker

Nathan Geer 氏 (Utah State University)

Title

**
Re-normalized Link invariants
**

Abstract

In the last few years, C. Blanchet, F. Costantino, B. Patureau, N. Reshetikhin, V. Turaev and myself (in various collaborations) have developed a theory of renormalized quantum invariants of links and 3-manifolds which lead to TQFTs. This talk will start out by giving an overview of this work. In the second part of the talk I will discuss the renormalized quantum invariants of links coming from quantized sl(2) at a root of unity. These link invariants contain Kashaev's quantum dilogarithm invariants of knots, the Akutsu-Deguchi-Ohtsuki invariant of links and the multi-variable Alexander Polynomial. Moreover, these re-normalized invariants of knots are meromorphic functions whose residues are closely related to the the colored Jones polynomials.

Date

2014年3月7日(金) 15:00 --

Room

京都大学 数理解析研究所 111号室

Speaker

清水 達郎 氏（東大数理）

Title

**
An invariant of rational homology 3-spheres via vector fields
**

Abstract

In this talk, we define an invariant of rational homology 3-spheres with values in a space $\mathcal A(\emptyset)$ of Jacobi diagrams by using vector fields. The construction of our invariant is a generalization of both that of the Kontsevich-Kuperberg- Thurston invariant $z^{KKT}$ and that of Fukaya and Watanabe's Morse homotopy invariant $z^{FW}$. As an application of our invariant, we prove that $z^{KKT}=z^{FW}$ for rational homology 3-spheres.

Date

2011年2月17日(木) 15:00〜

Room

京都大学 数理解析研究所 402号室

Speaker

Jean-Baptiste Meilhan 氏 (Universite Grenoble 1)

Title

**
Torelli group and equivalence relations for homology cylinders
(joint work with G. Massuyeau)
**

Abstract

Two 3-manifolds are called Y_k-equivalent if one can be obtained from the other by "twisting" an embedded surface by an element of the k-th term of the lower central series of its Torelli group. The J_k-equivalence relation is defined similarly, using the Johnson filtration instead of the lower central series. In this talk, we shall consider these equivalence relation among homology cylinders over a given surface S, which are 3-manifolds homologically equivalent to S \times [0,1]. We classify these equivalence relations, for k \le 3, using several classical invariants. This provides generalizations of results of W.Pitsch and S.Morita on the structure of integral homology spheres and the Casson invariant.

Date

2011年 1月25日(火) 15:00〜

Room

京都大学 数理解析研究所 110号室

Speaker

Sergei Duzhin 氏 (Steklov Mathematical Institute)

Title

**
A formula for the HOMFLY polynomial of rational links
(joint work with Mikhail Shkolnikov)
**

Abstract

We give an explicit formula for the HOMFLY polynomial of a rational link (in particular, a knot) in terms of a special continued fraction for the rational number that defines the given link.

Date

2010年6月10日(木) 15:00〜

Room

京都大学 数理解析研究所 402号室

Speaker

David Farris 氏 (University of California, Berkeley)

Title

**
The embedded contact homology of circle bundles over Riemann surfaces
**

Abstract

Embedded contact homology is a topological invariant of three-manifolds which is defined by choosing a contact structure on the manifold and studying pseudoholomorphic curves in the symplectization of the contact 3-manifold. We compute this invariant for circle bundles over Riemann surfaces (prequantization spaces), a case where pseudoholomorphic curves can be concretely understood as meromorphic sections of a holomorphic line bundle. We make use of domain-dependent almost complex structures to achieve transversality for moduli spaces of these curves.

Date

February 10 (Wednesday), 2010, 14:00-

Room

京都大学 理学部6号館 204講義室

Speaker

Mark Powell 氏（Edinburgh 大学）

Title

**
Knot concordance and twisted Blanchfield forms
**

Abstract

In this talk I will recall the notion of knot concordance as defined by Fox and Milnor, which asks whether a knot in S3 bounds a disk in 4 space D4. The work of Casson and Gordon involved a two stage obstruction theory which depends on the intersection form of a 4-manifold. This has been generalised by the work of Cochran-Orr-Teichner. I shall discuss an obstruction theory which is intrinsically 3-dimensional, using Blanchfield linking forms with coefficients twisted using metabelian representations of the knot group. These linking forms obstruct null-concordance. We then describe an algorithm to construct the symmetric chain complex of the universal cover of a knot exterior, and then use this to make calculations of the twisted Blanchfield forms.

Date

December 1 (Tuesday), 11:00-

Room

RIMS Annex (Research Building No.4), Room 307 (総合研究4号館307号室)

Speaker

Oliver Dasbach 氏 (Louisiana State University)

Title

**
On the Turaev surface and its applications
**

Abstract

Turaev constructed for each link diagram an embedded, unknotted surface on which the link projects alternatingly. We will show how the Jones polynomial can be computed from this surface. As a corollary we will get an interesting formula for the determinant of a knot. Furthermore, we will give applications to the study of Khovanov and Ozsvath-Szabo knot homologies.

Date

June 11 (Thur), 15:00-

Room

RIMS, Room 402

Speaker

Ivan Izmestiev 氏 (Technische Universitat Berlin)

Title

**
Hyperbolic cusps with convex polyhedral boundary
**

Abstract

Consider a hyperbolic cusp bounded by a locally convex piecewise geodesic
surface. The intrinsic metric of the boundary is a hyperbolic cone-metric
with cone angles less than 2\pi. We prove the converse: every hyperbolic
cone-metric on the torus with cone angles less than 2\pi can be realized
as the boundary of a convex hyperbolic cusp.

This theorem is similar to the Alexandrov's characterization of convex
polyhedra. In fact, both of them are special cases of a general statement
on convex realizations of surfaces with cone-metrics.

The proof is based on the variational properties of the discrete
Hilbert-Einstein functional.

This is a joint work with Francois Fillastre.

Date

2007年11月6日(木) 15:00 - 16:30

Room

数理解析研究所402号室

Speaker

Gwénaël Massuyeau (CNRS - Louis Pasteur University, Strasbourg)

Title

**
An infinitesimal version of Morita's homomorphisms
**

Abstract

Let S be a compact, connected, oriented surface with one boundary
component, and let I(S) be the Torelli group of S. For each integer
k>0, I(S) acts in the natural way on the k-th nilpotent quotient of
the fundamental group P(S) of S and, by definition, the kernel of this
action is the k-th term of the Johnson filtration of I(S). The Johnson
filtration being separated, one can study the Torelli group by
approaching P(S) by its successive nilpotent quotients. In this
context, Morita defined for each k>0 a group homomorphism from the
k-th term of the Johnson filtration to the third homology group of the
k-th nilpotent quotient of P(S).

In this talk, groups will be replaced by their Malcev Lie algebras,
which will lead to an ``infinitesimal'' version of Morita's
homomorphisms. Although equivalent to their originals, the
infinitesimal Morita homomorphisms seem more easy to use. Thus, we
will give a diagrammatic description for them and we will explain how
they can be computed. Finally, the infinitesimal Morita homomorphisms
will be connected to the diagrammatic representation of I(S) derived
from the LMO invariant of 3-manifolds.

Date

2007年8月23日(木), 午後2時〜4時(休憩時間30分を含む)

Room

京都大学数理解析研究所 102号室

Speaker

Yoav Rieck 氏 （University of Arkansas, USA）

Title

**
On the Heegaard genus of knot exteriors
**

Abstract

We will survey some of the authors' results about the behavior of Heegaard genus of knot exteriors under connected sum operation. As our main result we will prove that given integers g_i > 1 (i=1,...,n) there exist infinitely may knots K_i in S^3 so that g(E(K_i)) = g_i and g(E(K_1#...#K_n)) = g(E(K_1)) +...+ g(E(K_n)). This proves the existence of counterexamples to Morimoto's Conjecture.

Date

2007年4月12日(木), 14:00〜15:30 渡邉氏, 16:00〜17:30 石井氏

Room

京都大学 数理解析研究所 102号室

Speaker

渡邉 忠之 氏 （京大数理研、機関研究員） /石井 敦 氏 （京大数理研、日本学術振興会特別研究員）

Title

**
配置空間積分の特性類について
/
A bracket polynomial and TQFT for invariants of virtual links
**

Abstract

Maxim Kontsevichは、Chern--Simons摂動理論の高次元のアナロジーとして、
配置空間積分を用いて奇数次元の枠付きホモロジー球面をファイバーとする
ファイバーバンドル(C^\infty-smooth)の特性類を構成しました。この特性類は、
ファイバーが３次元の場合にはホモロジー３球面の位相不変量になり、
全てのQ値の有限型不変量（大槻不変量）を含む普遍的不変量であることが
Kuperberg--Thurstonにより示されています。しかし、ファイバーの次元が
３より大きい場合には、その性質は全く知られていませんでした。
この講演では、３より高い次元におけるKontsevichの特性類の非自明性に関して
得られた結果を紹介する予定です。またそれの系として、Casson不変量の高次元化
と思われる非自明な不変量が得られることなども時間があれば話す予定です。

仮想結び目理論において、ブラケット多項式を用いた不変量の構成方法と
オペレーター不変量の構成方法がどのように働くか（働かないか）を、
宮澤多項式を例にとって、古典的な場合との違いを説明します。

Date

September 28-29, 各日とも 14:00-17:00, 2006

Room

RIMS Room 402 (Sep. 28), Room 115 (Sep. 29), Kyoto University

Speaker

Sergei Duzhin 氏 / (Steklov Institute)

Title

**
Detecting the link orientation
/
Estimation of crossing numbers of knots**

Abstract

I will speak about the problem of detecting the orientation
of knots and links, i.e. finding the invariants that take distinct values
on two links differing only by an inversion.

The first result in this direction is a classical theorem of Trotter
who proved that the pretzel knot $P_{3,5,7}$ is not equivalent to its inverse.
Trotter and some subsequent authors used homomorphisms of the knot group
to study the invertibility.
It is known that knot polynomials obtained by the Reshetikhin--Turaev
procedure do not feel the orientation.
Finite type (Vassiliev) knot invariants are strictly stronger than quantum
invariants, and there is an important problem if these can tell a knot from
its inverse.
This problem is open until now.

For links with more than one component the corresponding problem
is partially solved, namely, a positive answer was obtained for closed links
with 6 or more components (X.-S.Lin) and for string links with 2 components
(S.Duzhin--M.Karev).

I will give a review of the known results on the problem and then speak
about an attempt to solve it for closed 2-component links using
the invariants with values in the necklace algebra.

Another approach that might lead to orientation-detecting invariants is
the categorification of the $gl_n$ Lie algebra weight system. Indeed, the
weight system with values in the center of the universal enveloping algebra
for $gl_n$ is a function on the space of Jacobi diagrams which is given by the
alternating sum over a cube of resolutions of the triple points of a
diagram. One may therefore try to apply the general scheme of
categorification due to Khovanov (through Frobenius algebras) and Bar-Natan
(through canopolies) -- in a hope to split the big alternating sum
into several smaller sums each of which gives a weight system,
whose totality is a stronger invariant than the original weight system.
This is an unfinished project, and I will speak about it in a hope to find
eventual collaborators among the listeners.

Date

July 13 (Thu), 2006, 15:00-

Room

Room 402, RIMS, Kyoto University

Speaker

Alexander Stoimenow 氏 / （京大数理研 COE研究員）

Title

**
結び目の交点数の評価
/
Estimation of crossing numbers of knots**

Abstract

5月10日の談話会で講演した内容の発展として、結び目図式から
結び目の交点数を決定したり評価したりする話題を中心にお話します。
とくに Lickorish-Thistlethwaite により定義された
semiadequate 絡み目とその部分クラスに対する交点数の評価を紹介し、
さらに、時間に余裕があった場合は、
それらの Jones 多項式の非自明性についてもお話したいとおもいます。

I intend to talk about problems related to determination and
estimation of crossing numbers of semiadequate knots, as defined
by Lickorish-Thistlethwaite, and some of their subclasses.
If time permits, I will discuss the relation to the
non-triviality of their Jones polynomial.

Speaker

14:30--15:30 Gwenael Massuyeau氏

16:00--17:00 高瀬 将道氏

Date

April 20 (Thu), 2006, 14:30-17:00

Room

Room 402, RIMS, Kyoto University

Speaker

Gwenael Massuyeau氏 （CNRS - Louis Pasteur University, Strasbourg / 日本学術振興会外国人特別研究員、京大数理研）

Title

**
Some finiteness properties for the Reidemeister-Turaev torsion of three-manifolds.**

Abstract

The Reidemeister-Turaev torsion is an invariant of a closed oriented three-dimensional manifold equipped with an Euler structure, with values in the ring of quotients of the group ring of the first homology group. We will prove that its reductions by powers of the augmentation ideal are finite-type invariants in the sense of M. Goussarov and K. Habiro. For this, we will start off by explaining how their theory of finite-type invariants can be refined to take into account Euler structures (which is a joint work with F. Deloup).

Speaker

高瀬 将道氏 (京大数理研 機関研究員)

Title

**
Homology 3-spheres in codimension three**

Abstract

For smooth embeddings of an integral homology 3-sphere in the 6-sphere, we define an integer invariant in terms of their Seifert surfaces. Our invariant gives a bijection between the set of smooth isotopy classes of such embeddings and the integers; and besides, gives rise to a complete invariant for homology cobordism classes of all embeddings of homology 3-spheres in the 6-sphere. As a consequence, we show that two embeddings of an oriented integral homology 3-sphere in the 6-sphere are isotopic if and only if they are homology cobordant. We also relate our invariant to the Rohlin invariant and accordingly characterise those embeddings which are compressible into the 5-sphere.