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

Date

2022年6月30日(木) 14:30--15:30

Room

Speaker

Andrew Kricker 氏 (Nanyang Technological University)

Title

On the asymptotics of the Garoufalidis-Kashaev meromorphic 3D index

Abstract

This talk will introduce arXiv:2109.05355, which is joint work with Craig Hodgson and Rafael Siejakowski. Garoufalidis and Kashaev have defined a fascinating topological invariant which associates to a 3-manifold with toroidal boundary a meromorphic function of two complex variables. It is defined from an ideal triangulation by a state-integral where a state is an element of S^1 assigned to every edge of the triangulation and the integrand is product of quantum dilogarithms obtained from the combinatorics of the triangulation. This invariant is a sort of generating function for the q-series 3D-index of Dimofte, Gaiotto and Gukov, and in fact proves topological invariance of it. We have studied a certain asymptotic limit of this function at the origin as the quantum parameter q approaches 1. Based on numerical investigations and integral heuristics we propose a conjecture for this asymptotic expansion involving surprisingly rich structure from the geometry of the manifold and its collection of boundary parabolic PSL(2,C)-representations. We prove a number of theorems about the expressions we obtain. A key structure we develop for this analysis is the concept of an S^1-valued angle structure which was introduced by Feng Luo.

Organizers T. Ohtsuki

Date

2020年2月6日(木) 14:30--15:30

Room

Speaker

Delphine Moussard 氏 (Aix-Marseille University)

Title

A splicing formula for the LMO invariant

Abstract

We will see a splicing formula'' for the LMO invariant of Le, Murakami and Ohtsuki. Specifically, if a rational homology 3-sphere M is obtained by gluing the exteriors of two framed knots in rational homology 3-spheres, the formula expresses the LMO invariant of M in terms of the Kontsevich-LMO invariants of the two knots. This is a joint work with Gwénaël Massuyeau.

Organizers T. Ohtsuki, K. Habiro

Date

2019年11月14日(木) 14:30--15:30

Room

Speaker

Gwenael Massuyeau 氏 (University of Burgundy)

Title

Generalized Dehn twists on surfaces and surgeries in 3-manifolds

Abstract

(Joint work with Yusuke Kuno.) Given an oriented surface S and a simple closed curve C in S, the "Dehn twist" along C is the homeomorphism of S defined by "twisting" S around C by a full twist. If the curve C is not simple, this transformation of S does not make sense anymore, but one can consider two possible generalizations: one possibility is to use the homotopy intersection form of S to "simulate" the action of a Dehn twist on the (Malcev completion of) the fundamental group of S; another possibility is to view C as a curve on the top boundary of the cylinder S \times [0,1], to push it arbitrarily into the interior so as to obtain, by surgery along the resulting knot, a new 3-manifold. In this talk, we will relate two those possible generalizations of a Dehn twist and we will give explicit formulas using a "symplectic expansion" of the fundamental group of S.

Organizers T. Ohtsuki, K. Habiro

Date

2019年10月17日(木) 14:30--15:30

Room

Speaker

Anderson Vera 氏 (RIMS, Kyoto University/JSPS)

Title

Alternative Johnson homomorphisms and the LMO functor

Abstract

One of the main objects associated to a surface S is its mapping class group MCG(S). By considering the action of MCG(S) on the fundamental group of S, it is possible to define different filtrations of MCG(S) together with some homomorphisms on each term of the filtration. In this talk we present our results concerning a filtration of MCG(S), "the alternative Johnson filtration", recently introduced by Habiro-Massuyeau and whose definition involves a handlebody bounded by S. We show the relationship between the "alternative Johnson homomorphisms" and the functorial extension of the Le-Murakami-Ohtsuki (LMO) invariant of 3-manifolds.

Organizers T. Ohtsuki, K. Habiro

Date

2019年7月9日(火) 14:30--15:30

Room

Speaker

Fiona Torzewska 氏 (University of Leeds)

Title

Using model categories to construct topological quantum field theories

Abstract

We are able to use Strom's model category structure on the category of topological spaces to construct a very general class of TQFTs, which can be applied to a variety of source categories. This uses ideas from Yetter's 'TQFTs assoiated to finite groups' and from Quinn's 'Finite total homotopy TQFT'.
In this talk I will outline the construction, highlighting why model categories appear to be the right setting for our work. I will also provide plenty of examples and show how explicit calculation is possible. No previous knowledge of model categories will be required.

Organizers T. Ohtsuki, K. Habiro

Date

2019年2月19日(火) 14:30--15:30

Room

Speaker

Jørgen Ellegaard Andersen 氏 (QGM, Aarhus University)

Title

The new formulation of the Teichmüller TQFT

Abstract

A few years ago we define the Teichmüller TQFT, which is based on a deformation of quantum Teichmüller theory. In its original formulation there was a certain homology assumption needed to guarantee finiteness of the involved integrals in the definition of that TQFT. In our new formulation, this assumption is no longer needed. The talk will be focused on our new formulation, its generalisations and if time permits the relation to the old formulation. The talk is based on joint work with Rinat Kashaev.

Organizers T. Ohtsuki, K. Habiro

Date

2018年11月22日(木) 14時30分 --

Room

Speaker

Gaetan Borot 氏 (Max Planck Institute for Mathematics)

Title

Mapping class group invariants via geometric recursion

Abstract

I will describe the geometric recursion (GR) and its first applications. Its aim is to construct mapping class group invariant quantities for surfaces of all topologies from a small amount of initial data, via excision/glueing of pairs of pants, and therefore by induction on the Euler characteristic. Although GR is more general, I will focus on the case where it takes values in the space of continuous functions over Teichmuller space of bordered surfaces. The GR functions are designed in such a way that, when integrated against the Weil-Petersson volume form, the result satisfies a topological recursion. For instance, Mirzakhani identities express the constant function 1 on Teichmuller spaces as a result of GR, and after integration one obtains a topological recursion for the Weil-Petersson volumes. I will present a family of generalizations of Mirzakhani identities involving statistics of hyperbolic lengths of simple multicurves, and give other examples of potential applications of GR. Based on a joint work with Jorgen Ellegaard Andersen and Nicolas Orantin.

Organizers T. Ohtsuki, K. Habiro

Date

2018年6月28日(木) 14時30分 --

Room

Speaker

Moussard Delphine 氏 (RIMS, Kyoto University / JSPS research fellow)

Title

2-knots with factorized Alexander polynomial

Abstract

For knots in the 3-sphere, it is well-known that the Alexander polynomial of a ribbon knot factorizes as f(t)f(1/t) for some polynomial f(t). For 2-knots, i.e. embeddings of a 2-sphere in the 4-sphere, the Alexander polynomial of a ribbon 2-knot is not even symmetric in general. Via an alternative notion of ribbon 2-knots, we give a topological condition on a 2-knot for recovering the factorization of the Alexander polynomial. This is a joint work with Emmanuel Wagner.

Organizers T. Ohtsuki, K. Habiro

Date

2018年6月21日(木) 14時30分 --

Room

Speaker

Zhongtao Wu 氏 (The Chinese University of Hong Kong)

Title

An Alexander polynomial for graphs

Abstract

Using Alexander modules, one can define a polynomial invariant for a certain class of graphs with a balanced coloring. We will give different interpretations of this polynomial by Kauffman state formula and MOY relations.
Moreover, we propose a potential categorification for the invariant. This is a joint work with Yuanyuan Bao.

Organizers T. Ohtsuki, K. Habiro

Date

2018年4月26日(木) 14時30分 --

Room

Speaker

Title

A_2 colored polynomials of rigid vertex graphs

Abstract

The Kauffman-Vogel polynomials are three variable polynomial invariants of 4-valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented 4-valent rigid vertex graphs was given by using the Kauffman bracket. We define an A_2 version of the one-variable Kauffman-Vogel polynomial for 4-valent rigid vertex graphs by using the A_2 bracket.

Organizers T. Ohtsuki, K. Habiro

Date

2018年4月12日(木) 14時00分 --

Room

Speaker

Title

Formulas for the action of Dehn twists on skein modules and their applications

Abstract

We give explicit formulaｓ for the action of the Dehn twist along a simple closed curve of a surface on the completed Kauffman bracket skein modules and the completed HOMFLY-PT type skein modules of the surface. As an application, using this formula, we construct some invariants for an integral homology 3-sphere.

Organizers T. Ohtsuki, K. Habiro

Date

2018年1月11日(木) 14時30分 --

Room

Speaker

Title

On Goussarov-Polyak-Viro's finite type invariants and a virtual $C_n$-equivalence on long virtual knots

Abstract

A $C_n$-move is a family of local moves on knots, which gives a topological characterization of finite type invariants of knots. On the other hand, Goussarov-Polyak-Viro defined finite type invariants and $n$-equivalence on (long) virtual knots. They mentioned that the value of a finite type invariant of degree less than or equal to $n$ depends only on the $n$-equivalence class of (long) virtual knots.
In this talk, we extend the $C_n$-move to (long) virtual knots by using the lower central series of the pure virtual braid, and we define a virtual $C_n$-equivalence, which is generated by this extended $C_n$-moves. We then prove that the virtual $C_n$-equivalence coincides with $(n-1)$-equivalence on long virtual knots.

Organizers T. Ohtsuki, K. Habiro

Date

2017年10月19日(木) 14時30分 --

Room

Speaker

Andrew Kricker 氏 (Nanyang Technological University)

Title

What is the l^2-Alexander invariant and how can we approximate it?

Abstract

The classical Alexander polynomial marks the beginning of mathematical knot theory. The l^2-Alexander invariant is a recent variation due to Li-Zhang (2006) and Dubois-Friedl-Luck (2014) where you twist the classical construction by the infinite-dimensional representation consisting of square-summable series of elements of the fundamental group. This invariant has remarkable properties - such as the fact that it determines the volume of a hyperbolic knot, determines the Thurston norm, and detects infinitely many knots. In this talk I'll introduce this invariant and survey what is known and unknown about it. It is a very strong invariant, but also is almost impossible to calculate. I'll also talk about some ongoing projects investigating its approximation.

Date

2016年11月24日(木) 14:30 -- 15:30

Room

Speaker

Title

Heegaard surfaces and singularities of product maps

Abstract

The classification of Heegaard surfaces is a general problem in 3-manifold topology. Some recent progress on this problem has been made with a technique based on the singularities of the product map of two smooth functions. Further progress seems to depend on a special singularity theory for product maps. In this talk, I would like to survey relevant results, pending problems and my approach.

Date

2016年10月20日(木) 14:30 -- 15:30

Room

Speaker

Moussard Delphine 氏 (RIMS, Kyoto University / JSPS research fellow)

Title

Equivariant triple intersections

Abstract

We define an invariant of null-homologous knots in rational homology 3-spheres by means of equivariant triple intersections of surfaces in the infinite cyclic covering of the complement of the knot. The invariant is a map on the triple tensor product of the Alexander module of the knot. We compute the variation of this invariant under null Borromean surgeries, and describe the set of all maps obtained as such invariants.

Date

2016年6月22日(水) 15:00 -- 16:00

Room

Speaker

Title

カウフマン・ブラケット・スケイン代数による整係数ホモロジー球面の不変量

Abstract

Torelli群から完備化されたカウフマン・ブラケット・スケイン代数への 埋め込みにより、トレリ群とカウフマン・ブラケット・スケイン代数の 新たな関係性を発見することができた。この埋め込みとHeegaard splitting を用いて整係数ホモロジー球面の不変量を構成することができた。この 不変量は Q[[A+1]] に値を持つ。さらにmod (A+1)^n で見たとき 位数 n の有限型不変量になる。とくに A+1 の係数はCasson invariantとなる。 この不変量の構成の仕方と不変量になっている証明の概略を紹介する。

Date

2015年12月16日(水) 14:30 -- 15:30

Room

Speaker

Nathan Geer 氏 (Utah State University)

Title

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

Speaker

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

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

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

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

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

2007年8月23日(木), 午後2時～4時(休憩時間30分を含む)

Room

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

Speaker

Title

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。

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

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

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.

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