On non-uniformly simple groups (非一様完全な単純群について)
Hiroki Kodama
(University of Tokyo)
Abstract: Suppose G is a simple group. For any nontrivial elements g and h of G, g can be written as a finite product of conjugates of h or the inverse of h. G is called uniformly simple if the length of such an expression is uniformly bounded. We show that the infinite alternating group is non-uniformly simple and evaluate how the length of such an expression is unbounded.
G を単純群とし G の非自明な元 g, h をとると、g は h の共役元 および h の逆元の共役元の有限個の積として書ける。この積の表示の長さが上から一様に押さえられるとき、G は一様完全であるという。 本講演では無限交代群に対して積の表示の長さを評価し、とくに無限交代群が一様完全ではない完全群であることを示す。
Heegaard Floer homology and strong L-spaces
Takuya Usui
(University of Tokyo)
Abstract: Heegaard-Floerホモロジーとは有向閉3次元多様体のある位相不変量である。 また、L空間とはHeegaard-Floerホモロジーが最も良い形になるような、 有理ホモロジー球面である。本講演では、Heegaard-Floerホモロジー の境界準同型が消えるような、特別なL空間（strong L-spaces、強L空間）につ いて考察する。 強L空間が組み合わせ的に扱いやすい対象となることを説明し、 Heegaard分解の種数が3以下の場合における分類定理を証明する。
No Seminar
Quantum invariants for handlebody-knots
Atsuhiko Mizusawa
(Waseda university)
Abstract: A handlebody-knot is an embedding of a handlebody into a 3-manifold. A handlebody-knot is represented by a trivalent spatial graph that have a regular neighborhood which is ambient isotopic to the handlebody-knot. In this talk, we define quantum invariants for handlebody-knots in a 3-sphere using Yokota's invariants, which are the invariants for spatial graphs. We also see properties of the invariants and examples of calculations.
Winter Break, No Seminar
Meridional destabilizing number and connected sums of knots
Toshio Saito
(Joetsu University of Education)
Abstract: From a viewpoint of Heegaard theory, we have two types of natural positions of knots in closed orientable 3-manifolds: a bridge position with respect to a Heegaard surface, and a core position of a handlebody bounded by a Heegaard surface. The latter has a close connection to Heegaard genus (or tunnel number) of knots. Meridional destabilizing number, which is defined by considering such two positions, will be introduced in this talk. We could say this together with Heegaard genus gives a binary complexity of knots. We will then discuss its behavior for composite knots.
On the twisted Alexander polynomial for once-punctured torus bundles
Yoshikazu Yamaguchi
(Tokyo Institute of Technology)
Abstract: I would like to explain a computation of twisted Alexander polynomials for once-punctured torus bundles over the circle. This computation is based on the expression of character varieties for once-punctured torus bundles as the fixed point sets in the character variety for the fiber under the actions induced by the monodromies. We will see explicit computations of these actions induced by monodromies and discuss the twisted Alexander polynomial for once-punctured torus bundles.
On region crossing changes and a game
Ayaka Shimizu
(Osaka City University Advanced Mathematical Institute)
Abstract: A region crossing change at a region of a link diagram is defined to be the crossing changes at the crossing points on the boundary of the region. We show that the region crossing change on a knot diagram is an unknotting operation. We also discuss the region crossing change on link diagrams. As an application, we introduce the game "Region Select" which is a joint work with Akio Kawauchi and Kengo Kishimoto.
The knot concordance invariant and the band modification
Keiji Tagami
(Tokyo Institute of Technology)
Abstract: We call a knot invariant ν of the Livingston-Naik type if it is a smooth knot concordance invariant that bounds the 4-genus of a knot and determines the 4-genus of any positive torus knot. For example, the Ozsváth-Szabó invariant τ and the Rasmussen invariant s are of Livingston-Naik type. A Seifert surface for a knot, viewed as a disk with band attached, can be modified by removing one of the bands and reattaching it in the same place. The reattaching band may be twisted and knotted. We call this operation the band modification. In this talk, we introduce a relation between a Livingston-Naik type invariant and a band modification. This is a work of Livingston and Naik.
No seminar
A two dimensional lattice of knots by Cn-moves
Sumiko Horiuchi (joint work with Yoshiyuki Ohyama)
(Tokyo Woman’s Christian University)
Abstract: We consider a local move on a knot diagram, where we denote the local move by M. If two knots K1 and K2 are transformed into each other by a finite sequence of M-moves, the M-distance between K1 and K2 is the minimum number of times of M-moves needed to transform K1 into K2. A M-distance satisfies the axioms of distance.
A two dimensional lattice of knots by M-moves is the two dimensional lattice graph which satisfies the following : The vertex set consists of oriented knots and for any two vertices K1 and K2, the distance on the graph from K1 to K2 coincides with the M-distance between K1 and K2, where the distance on the graph means the number of edges of the shortest path which connects the two knots.
Local moves called Cn-moves are closely related to Vassiliev invariants. We obtain that for any given knot K, there is a two dimensional lattice of knots by Cn-moves (n≥ 4) with the vertex K.
No seminar
No seminar
Topologically slice knots with non-trivial Alexander
polynomial (2)
Yuanyuan Bao
(Tokyo Institute of Technology)
Abstract: This is the continuance of the talk given two weeks ago. We explain the calculation of Ozsváth and Szabó's correction terms in their paper.
On nanophrases and mu bar invariant
Yuka Kotorii
(Tokyo Institute of Technology)
Abstract: A word is a sequence of symbols, called letters, belonging to a given set, called alphabet. Turaev developed the theory of words based on the analogy with curves on the plane, knots in the 3-sphere, virtual knots, etc. Homotopy classes of nanophrases are combinatorial generalizations of virtual links.
Two link diagrams are link homotopic if one may be transformed into the other by a sequence of Reidemeister moves and self crossing changes. Milnor introduced an invariant under link homotopy called mu bar invariant. In this talk we extend link homotopy to nanophrases corresponding to virtual links and mu bar invariant to such nanophrases.
Topologically slice knots with non-trivial Alexander
polynomial (1)
Yuanyuan Bao
(Tokyo Institute of Technology)
Abstract: This talk is a report of a paper written by Hedden, Livingston and Ruberman. They proved that there are infinitely many topologically slice knots which are not smoothly concordant to any knots with trivial Alexander polynomial. Note that knots with trivial Alexander polynomial are topologically slice knots. The tool they are using is Ozsváth and Szabó's correction terms.
After July 7th, there is no seminar until October.
Casson-Gordon invariants and correction terms in Heegaard Floer homology
Yuanyuan Bao
(Tokyo Institute of Technology)
Abstract: In this talk, we first review the definition of Casson-Gordon invariants of a knot and their famous properties for slice knots. Then we review some recent developments of knot concordance related to correction terms in Heegaard Floer homology. We put these two topics together since they bear some resemblance, which is of great interest. The talk is based on works due to Casson, Gordon, Grigsby, Ruberman, Strle, Manolescu, Owens and some other people.
Twisted Alexander polynomial of knots in finite cyclic branched covers of the 3-sphere
Yoshikazu Yamaguchi
(Tokyo Institute of Technology)
Abstract: When we consider a pair of a finite cyclic branched cover of the 3-sphere branched over a knot K and the preimage of K, we almost often have a null-homologous knot in a rational homology 3-sphere. Then we can express the twisted Alexander polynomial of such a knot in a rational homology 3-sphere, given as the branched set in a finite cyclic branched cover of the 3-sphere, by using the product of twisted Alexander polynomials of a knot in the 3-sphere.
In this talk, we give an exposition of this formula for twisted Alexander polynomial via an explicit example for the figure eight knot. If we have time, we will discuss recent work in progress on metabelian covers.
Introduction to Khovanov homology
Keiji Tagami
(Tokyo Institute of Technology)
Abstract: In 2000, M. Khovanov introduced a homology type invariant of links. For any link L, he constructed a graded cochain complex (C(L), d) whose graded Euler characteristic is the (unnormalized) Jones polynomial of L and whose homotopy class is an invariant of L. We call its homology groups Khovanov homology of L.
In this talk, I would like to introduce the definition of Khovanov homology and some properties. Moreover by using a long exact sequence we compute the Khovanov homologies of torus links for some homological degrees.
A new invariant for virtual knots and forbidden moves
AND
2 and 3-variations and finite type invariants of degree 2 and 3
Migiwa Sakurai
(Tokyo Woman's Christian University)
Abstract: The talk consists of two parts:
Part 1: It is known that any virtual knot can be deformed into a trivial knot by a finite sequence of forbidden moves. In this talk, we define a new invariant p(K) for virtual knots and give the difference of the values of p(K) between two knots which can be transformed into each other by a single forbidden move.
Part 2: M. Goussarov, M. Polyak, and O. Viro defined a finite type invariant for virtual knots. In this talk, we give the difference of the values of the finite type invariant of degree two and three between two knots which can be transformed into each other by a 2 and 3- variation, respectively.
Δ Y -exchanges and the Conway-Gordon theorems
Ryo Nikkuni
(Tokyo Woman's Christian University)
(joint work with K. Taniyama)
Abstract: Conway-Gordon proved that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this talk, we give a Conway-Gordon type theorem for any graph which is obtained from the complete graph on 6 or 7 vertices by a finite sequence of Δ Y -exchanges.
Constructions of maps with good singularities whose codimensions are -1 between low dimensional manifolds
Naoki Kitazawa
(Tokyo Institute of Technology)
ABSTRACT: One of the methods of studying the topology of manifolds is to use functions which have good properties (about singularities); Morse theory. The theory has been useful. Recently, a generalization of Morse theory is actively studied; we take general Euclidian spaces or more generally, general manifolds as the target (Let the dimensions of the targets be lower than those of the sources.). In the theory maps which we generalize Morse functions, whose singularities are also good, are essential. So it is important to study about such maps including maps called stable maps. Stable maps exist plentiful when the pairs of dimensions are good (As a special case, pairs of low dimensions are good. See [1].). However, it is very difficult to construct examples of such good maps whose singularities are clear except trivial ones (See section 6 of [2], for example.), though examples are important.
In this talk, we introduce some operations on maps to construct new maps from a few ones. We introduce some examples of constructions of maps and applications. In this talk, we study mainly about maps whose pairs of dimensions are (3,2) or (4,3).(3,2) and (4,3) cases are higher versions of the (2,1) case, classical Morse theory on surfaces.References
Torelli group and Equivalence relations for homology cylinders
Jean-Baptiste Meilhan
(joint work with G. Massuyeau)
(Université Joseph Fourier)
ABSTRACT: Two 3-manifolds are called Yk-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 Jk-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 × [0,1]. We classify these equivalence relations, for k ≤ 3, using several classical invariants. These classification results provide generalizations of results of W.Pitsch and S.Morita on the structure of integral homology spheres and the Casson invariant.
No Seminar
Brunnian braids and Brunnian links
Fedor Duzhin
(Nanyang Technological University)
ABSTRACT: A Brunnian link is a link that becomes trivial after removing any of its components. Similarly, a Brunnian braid is a braid that becomes trivial after removing any of its strands. According to Alexander's theorem, any link can be obtained as a closure of some braid. Obviously, the closure of a Brunnian braid is a Brunnian link. However, some not every Brunnian link is the closure of a Brunnian braid. In this talk we'll discuss some ways of constructing Brunnian links that cannot be obtained as a closure of a Brunnian braid.
No Seminar
(The next seminar will be on January 6th 2011)
On the Conway potential function introduced by Kauffman
Masashi Sato
(Tokyo Institute of Technology)
ABSTRACT: In this talk we show two results about the Conway potential function which is known as the normalized multivariable Alexander polynomial. We first show that the Conway potential function introduced by Kauffman in Formal knot theory is indeed a link invariant. Next we show that Kauffman's potential function equals Hartley's potential function. We will prove it by using Murakami’s axioms for the multivariable Alexander polynomial.
Progress on solving the equivalence problem for flat virtual links
Andrew Gibson
(Tokyo Institute of Technology)
ABSTRACT: An n-component flat virtual link is a collection of n curves on a surface modulo homotopy and stable homeomorphism. Kadokami defined a method to determine whether two flat virtual links are equivalent or not, but a hole was found in the pro of of its correctness. Furthermore, there is a counter-example for the multi-component case. In this talk we will give a new proof that Kadokami's method is correct in the case of a single component. If there is time we will look at how we might co rrect Kadokami's method in the multi-component case. (Joint work with Teruhisa Kadokami of East China Normal University)
The clock number of a knot
Yukiko Abe
(Tokyo Institute of Technology)
ABSTRACT: This talk is about the clock number of a knot. First we define the clock number using the states of a knot which Kaffman defined in "Formal Knot Theory". Next we show that if K is a prime knot, its clock number is more than or equal to its crossing number. Finally we prove that its clock number is equal to its crossing number if and only if K is a two-bridge knot.
On the twisted Alexander polynomial for metabelian SL(2,C)-representations
Yoshikazu Yamaguchi
(Tokyo Institute of Technology)
ABSTRACT: In this talk, we compare the twisted Alexander polynomial of knots for SL(2,C)-metabelian representations with that for the composition of SL(2,C)-metabelian representations with the adjoint action. We can find the (classical) Alexander polynomial as a factor in the case of the twisted Alexander polynomial with the adjoint action, unlike the twisted Alexander polynomial for standard SL(2,C)-metabelian representations. We also discuss the characterization of metabelian representations in character varieties via the actions of finite cyclic g roups. It is the background on our computation result which has the Alexander polynomial as a factor in the twisted Alexander polynomial.
Seifert Surgery Network and chain links
Kimihiko Motegi
(Nihon University)
(joint with Arnaud Deruelle and Katura Miyazaki)
ABSTRACT: We have proposed a new approach in studying Dehn surgeries on knots in the 3-sphere yielding Seifert fiber spaces. Our basic idea is finding relationships among such surgeries. To describe relationships and get a global picture of Seifert surgeries, we introduced “seiferters” and the Seifert Surgery Network, a 1-dimensional complex whose vertices correspond to Seifert surgeries. In our study it is important to find paths in the network from Seifert surgeries on hyperbolic knots to those on a torus knots, which are regarded as sources of Seifert surgeries. In this talk we start with a brief review of the Seifert Surgery Network and then describe some relations between chain links and the Seifert Surgery Network.
A formula for the HOMFLY polynomial of rational links
Sergei Duzhin
(Waseda University)
ABSTRACT: We give an explicit formula for the HOMFLY polynomial of a rational link (in particular, knot) in terms of a special continued fraction for the rational number that defines the given link. The talk is based on the paper arXiv:1009.1800, joint with my student M.Shkolnikov.
On genaralized link-homotopy of links
Akira Yasuhara
(Tokyo Gakugei University)
ABSTRACT: Milnor defined link-homotopy which is an equivalence relation on links generated by self crossing change. Where self crossing is the crossing change whose strands belong to the same component. Habeger and Lin gave a classification on links up to link-homotopy. Murakami-Nakanishi and Murakami defined respectively delta-move and sharp-move that are known as genaralized crossing change.
In this talk, we consider self-sharp equivalence and self-delta equivalence as generalized link-homotopy, and give several results we have shown so far.This talk is based on joint works with Tetsuo Shibuya.
No seminar
The colored Jones polynomial, the Chern-Simons invariant and the Reidemeister
torsion
Hitoshi Murakami
(Tokyo Institute of Technology)
ABSTRACT: This talk is about a work in progress. We will try to show that for a real number u, the asymptotic behavior of the colored Jones polynomial of the figure-eight knot evaluated at exp((u+2*π*I)/N) gives the SL(2;C) Chern-Simons invariant and the twisted Reidemeister torsion associated with a representation from the fundamental group of the knot complement to SL(2;C) parameterized by u.
Unknotting operations and Heegaard Floer homology
Yuanyuan Bao
(Tokyo Institute of Technology)
ABSTRACT: Ozsvath and Szabo (Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary. Adv. Math. 173 (2003), no. 2, 179--261) defined an invariant called "correction term" for a rational homology three-sphere associated with a torsion spin^c-structure. They used this invariant for a given three-manifold to study the intersection form of a four-manifold which is bounded by the given three-manifold. These results were applied by Ozsvath and Szabo (Knots with unknotting number one and Heegaard Floer homology. Topology 44 (2005), no. 4, 705--745) to give an obstruction to unknotting a knot by a crossing change. Owens generalized their study and calculated the unknotting numbers of some other knots.
In this talk, we will review these works. After that, we show that the ideas involved in Ozsvath and Szabo's paper can also be used to study the H(2)-unknotting number of a knot. Precisely, we state an obstruction to unknotting a knot by adding a twisted band.
Transversely affine foliations
Naoki Kato
(University of Tokyo)
ABSTRACT: Transversely affine foliations are foliations which have transverse affine structures. One dimensional transversely affine foliations of two or three dimensional manifolds are classified By T. Nishimori and S. Matsumoto.
In this talk, we discuss a classification of codimension two transversely affine foliations of n-dimensional torus bundles over the circle.
An introduction to the sheet number of a surface-knot
Chikara Haruta
(University of Tokyo)
ABSTRACT: A connected surface smoothly embedded in R4 is called a surface-knot. In particular, if a surface-knot F is homeomorphic to a 2-sphere or a projective plane, then it is called a S2-knot or a P2-knot, respectively. The sheet number of a surface-knot is an invariant analogous to the crossing number of a 1-knot. M. Saito and S. Satoh proved some results concerning the sheet number of a S2-knot. In this talk, we will review the definition of the sheet number of a surface-knot, and a result by the author concerning the sheet number of a P2-knot.
Symmetric unions with identical knot Floer homology
Toshifumi Tanaka
(Gifu University)
ABSTRACT: A symmetric union is a generalization of a connected sum of a knot and its mirror image, and it is known to be an example of a ribbon knot. In this talk, I will give a pair of (non-two-bridge) symmetric unions with the same Khovanov homology and the same knot Floer homology, but different Alexander modules and explain how to calculate Khovanov homology and knot Floer homology by using their exact sequences. This is a joint work with Jae Choon Cha (POSTECH).
Factorization formulas of higher-order Alexander invariants for homological fibered knots
Takuya Sakasai
(Tokyo Institute of Technology)
ABSTRACT: This is a joint work with Hiroshi Goda (Tokyo University of Agriculture and Technology).
Homologically fibered knots are knots whose exteriors satisfy the same homological conditions as fibered knots. We focus on higher-order Alexander invariants for homologically fibered knots and see that they are generally factorized into two parts: Reidemeister torsions and Magnus matrices. We discuss the meaning of the factorization and give their applications to the topology of homological fibered knots.
An introduction to invariants of free knots
Andrew Gibson
(Tokyo Institute of Technology)
ABSTRACT: Free knots are virtual knots modulo crossing changes and the virtual switch operation. Equivalently, free knots can be defined as homotopy equivalence classes of Gauss words. Manturov and I independently showed that there are infinitely many free knots. This will be an introductory talk about some of Manturov's invariants for free knots.
Exceptional surgeries on alternating knots
Kazuhiro Ichihara
(Nihon University)
ABSTRACT: We will review results on Dehn surgeries yielding non-hyperbolic manifolds, called exceptional surgeries, on hyperbolic alternating knots in the 3-sphere. Also we will report on recent related works by the author including joint works with I. D. Jong and S. Mizushima.
Moduli of Bridgeland semistable objects on the projective plane and its wall-crossing
Ryo Ohkawa
(Tokyo Institute of Technology)
ABSTRACT: We introduce the notion "stability conditions on triangulated categories", which is introduced by Bridgeland, and call them Bridgeland stability. Comparing classical Gieseker stability with Bridgeland stability on the bounded derived category of coherent sheaves on the projective plane, we construct moduli spaces of semistable torsion free sheaves on the projective plane as moduli spaces of semistable representations over certain finite dimensional algebras. As an application we describe flip diagrams of the moduli spaces when the first Chern class is equal to 1.
Extremal coefficients in the Homfly polynomial
Tamas Kalman
(Tokyo Institute of Technology)
ABSTRACT: I will explain a construction that associates to a given bipartite graph a polytope P, and to P a pair of one-variable polynomials I and E. I will also discuss the notion of `trinity' for a triple of planar bipartite graphs and how their polytopes and polynomials relate. Now if the graph G is the Seifert graph of an alternating link projection (ie., if it is bipartite and planar), then, conjecturally, the corresponding I is a valuation of the link's Homfly polynomial. On the other hand, in joint work with Juhasz and Rasmussen, we show that P is the spin-c polytope of a natural sutured manifold. Thus, once the conjecture is proven, we get a concrete connection between Heegaard Floer homology and the Homfly polynomial.
A bicomplex of Khovanov homology for colored Jones polynomial
Noboru Ito
(Waseda University)
ABSTRACT: We discuss the existence of a bicomplex which is a Khovanov-type complex associated with categorification of colored Jones polynomial. This is an answer to the question proposed by A. Beliakova and S. Wehrli. Then the second term of the spectral sequence of the bicomplex corresponds to the Khovanov-type homology group. In this talk, we devote ourselves to the definition of this bicomplex.
Goussarov-Polyak-Viro finite type invariants of nanowords and nanophrases
Andrew Gibson
(Tokyo Institute of Technology)
(joint work with Noboru Ito of Waseda University)
ABSTRACT: Nanowords and nanophrases can be viewed as combinatorial generalizations of virtual knot and link diagrams. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links using the virtualization operation. In this talk, we generalize their definition to nanowords and nanophrases and give several examples of finite type invariants.
On the knot Floer homology of a class of satellite knots
Yuanyuan Bao
(Tokyo Institute of Technology)
ABSTRACT:
In this talk, we discuss the knot Floer homology of a class of satellite knots.
We start from a brief recollection of the knot Floer homology theory. Then the research
motivation is given. Afterwards the talk goes as follows:
(1) The Heegaard diagram of the satellite knots.
(2) The Alexander grading and the homology at the top grading.
(3) An application to the Seifert genus.
For simplicity, we explain everything through an example.
３次元多様体の三角形分割とnormal surface について
小林 雅子 氏
ABSTRACT: 三次元多様体に埋め込まれた二次元多様体を分析する道具としてnormal surface 理論 を挙げることができるが，この理論を適応するためには三次元多様体の三角形分割をでき るだけ簡単なものにすることが有効である．
本講演ではいくつかの多様体についての三角形分割の方法とその分割についてのnormal surface の埋め込まれ方について議論する．
この講演は，文部科学省技術振興調整費「理工系女性研究者プロモーションプログラム」 の助成を受けています．
また，16時15分から引き続き「『女性』数学者と呼ばれることへの疑問」と題した講演 もお願いしています．17時からは座談会も予定しています．『女』医，『女流』作家のよ うにことさら女性であることを強調する言葉遣いの背後に潜む「男性との区分」について いっしょに考えてみましょう．
http://www.leap-titech-ac.jp/topics/event/201002/20100204000000246.html
Multivariable twisted Alexander polynomial for hyperbolic link exteriors
Yoshikazu Yamaguchi
(University of Tokyo)
ABSTRACT: There is a relation between the twisted Alexander invariant of a hyperbolic knot and the hyperbolic torsion of the knot exterior, which is called derivative formula. This relation can be extended to link exteriors whose linking numbers for any two components are all zero, like as the Whitehead link. We will explicitly describe this relation in the case of the Whitehead link and also see an application to the hyperbolic torsion of twist knot exteriors.
On the Cn-distance and Vassiliev invariants
Sumiko Horiuchi
(Tokyo Woman's Christian University)
(joint work with Yoshiyuki Ohyama)
ABSTRACT: M. N. Goussarov and K. Habiro showed independently that a local move called a C_n-move is closely related to Vassiliev invariants. A C_n-distance between two knots K and L, denoted by d_{C_n}(K,L), is the minimum number of C_n-moves needed to transform K into L. Let p and q be natural numbers with p>q≧1. In this talk, we show that for any pair of knots K_1 and K_2 with d_{C_n}(K_{1},K_{2})=p and for any given natural number m, there exist infinitely many knots J_{i} (i=1,2,…) such that d_{C_n}(K_{1},J_{i})=q and d_{C_n}(J_{i},K_{2})=p-q, and they have the same Vassiliev invariants of order less than or equal to m. In the case that n=1 or 2, the knots J_{i} (i=1,2,…) satisfy more conditions.
Introduction to surface links and surface braids
Inasa Nakamura
(University of Tokyo)
ABSTRACT: Closed 2-manifolds locally flatly embedded in the Euclidean 4-space are called surface links. Similarly to the 1-dimensional knot theory, there is a notion of surface braids, and Kamada showed that 2-dimensional Alexander's theorem holds for oriented surface links. In this talk we will introduce charts, which represent simple surface braids. By 2-dimensional Alexander's theorem, any oriented surface link is described by a chart. Moreover we will give the definition of C-moves, which are equivalence relations on charts of the same degree, and give several examples.
３次元多様体のcovering表示とTuraev-Viro不変量
畠中英里 氏
(東京農工大学 女性未来育成機構 講師)
ABSTRACT: 3次元多様体のcovering表示とは，その３次元多様体から３次元球面へのsimple な分岐被覆を与えた時の，底空間の分岐集合としてとれる絡み目のことである．この表示方法は近年Bobtcheva-Piergalliniによって与えられた．
本講演ではcovering表示を使って３次元多様体のTuraev-Viro不変量を再構成することを試みる．さらにそのような不変量の構成法を一般化することについて考える．
この講演は，文部科学省技術振興調整費「理工系女性研究者プロモーションプログラム」の助成を受けています．
また，11時15分から引き続き「女性研究者としての私のワークライフバランス」と題して，東京農工大学における男女共同参画への取り組みとともに，子育てをしながらの研究体験などについてお話しいただきます．12時からは座談会も予定しています．次のウェブサイトもご参照ください．
http://www.leap-titech-ac.jp/topics/event/200912/20091203000000206.html
On the Turaev surface and its applications
Oliver Dasbach
(Louisiana State University)
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. Furthermore, we will give applications to the study of Khovanov and Ozsvath-Szabo knot homologies.
On generalization of homotopy of words and phrases
Tomonori Fukunaga
(Hokkaido University)
ABSTRACT: V. Turaev introduced the theory of homotopy of words and phrases. This is combinatorial extension of the theory of virtual strings and virtual knots. In this talk, we generalize the notion of homotopy of words and phrases, and we give some geometric meanings of the generalized homotopy of words and phrases. Further, we introduce some generalized homotopy invariants for nanowords which is derived from S-homotopy invariants for nanophrases.
Local moves on link diagrams
Hitoshi Murakami
(Tokyo Institute of Technology)
ABSTRACT: Given two links, there exist a series of crossing changes that change one into the other if and only if they have the same number of components. Similarly there exist a series of Delta unknotting operations that change one into the other if and only if they have the same `link homology', where we do not use `link homology' in a modern sense like the Khovanov homology. I will introduce such local moves and discuss how to classify links up to them.
On the coarse geometry of Teichmueller space
Kenneth J. Shackleton
(University of Tokyo IPMU)
ABSTRACT: We discuss the synthetic geometry of the pants graph in comparison with the Weil-Petersson metric, whose geometry the pants graph coarsely models following work of Brock. We also restrict our attention to the 5-holed sphere, studying the Gromov bordification of the pants graph and the dynamics of pseudo-Anosov mapping classes.
Spin TQFT
Jerome Petit
(Tokyo Institute of Technology)
ABSTRACT: I will give an introduction to spin TQFTs. Spin TQFTs can be viewed as a TQFT with an additional data (spin structure). I will give different construction of spin TQFTs and spin invariants of 3-manifolds.
Introduction to Definitions of Heegaard (Link) Floer Homology
Yuanyuan Bao
(Tokyo Institute of Technology)
ABSTRACT: This talk is based on a series of papers by P. Ozsváth and Z. Szabó and their collaborators, where they defined and studied the now named Heegaard (Link) Floer homology for oriented closed 3-manifolds and null-homologous links in a 3-manifold.
We will review the definitions for these homologies in the first half part of this talk. Even though the definitions are generally analytic and hard to apply to calculation, for special setting, however, some analytic elements can be converted to combinatorial data. In the second half part, we will mention this conversion, and in the end take lens spaces and trefoil knot as examples to show the calculation.
Introduction to the unknotting number of a knot
Hitoshi Murakami
(Tokyo Institute of Technology)
ABSTRACT: I will give an introduction to the unknotting number of a knot. Starting with the definition, I will describe its relation to the Alexander invariant of knots and the linking form on three-manifolds.
Homotopies of nanowords and their factorizations
Andrew Gibson
(Tokyo Institute of Technology)
ABSTRACT: Nanowords were defined by Turaev and can be viewed as a generalization of virtual knots. Turaev defined moves on nanowords which are parameterized by some data called a homotopy data triple. The moves define an equivalence relation on nanowords called homotopy. If we pick a different homotopy data triple, then we may get a different equivalence relation. In this talk we will define a product on homotopy data triples and introduce the concept of prime and composite homotopy data triples. Then we will show that homotopies given by composite homotopy data triples can be calculated in terms of the homotopies given by their prime factors.
A review of geometrization conjecture and the works of G. Perelman
Sergei Duzhin
(Steklov Mathematical Institute, Petersburg Division)
ABSTRACT: The relation between topology and metric geometry of 2-dimensional closed surfaces is well-known: each surface has a metric of constant curvature whose sign coincides with the sign of the Euler characteristic of the surface (positive for the sphere, zero for the torus and negative for the surfaces of genus greater than 1).
Around 1980 W. Thurston stated a conjecture that the situation with three-manifolds is similar, albeit much more complicated. He described eight homogeneous three-dimensional Riemannian geometries (three geometries of constant curvature and five homogeneous, but not isotropic) and claimed, with some grounds, that any compact 3-manifold can be split into pieces each of which admits one of the eight model geometries. The Thurston gemoetrization conjecture includes, as a particular case, the Poincare conjecture that any oriented connected 3-manifold with trivial fundamental group is homeomorphic to the 3-sphere.
During 25 years many mathematicians worked on the geometrization program. They obtained lots of results, but the conjecture itself sustained the attack. The elliptic and the hyperbolic cases presented the most difficulties.
In a series of three preprints dated 2002--2003, Grigory Perelman, building on the ideas of R. Hamilton, gave a complete solution to the problem of geometrization, investigating the evolution of the Riemannian manifold under the Ricci flow. His preprints were rather concise, so some details needed to be filled in, which was done by two independent groups of experts who published long texts on the subject by 2006.
In this talk, I will give an introduction into three-dimensional topology, describe the geometrization conjecture and say some words about the new ideas brought in by G. Perelman.
Conway polynomial as a function of combed braids via Magnus expansion
Sergei Duzhin
(Steklov Mathematical Institute, Petersburg Division)
ABSTRACT: The short-circuit closure of pure braids induces a map from 3-braids onto 2-bridge knots. Any Vassiliev invariant of knots thus becomes a finite-type invariant of 3-braids. There is a universal finite type invariant of pure braids given by the Magnus expansion. In this talk, we will explicitly describe the map from horizontal chord diagrams on 3 strands obtained by factoring the Conway polynomial through Magnus expansion. The result is described by a peculiar combinatorial map from ordered partitions of an integer into non-ordered partitions of the same integer. An open problem related to the arithmetical progressions in the set of Conway polynomials of 2-bridge knots will be stated. The topics of my further research include (1) generalization to braids with an arbitrary number of strands, (2) generalization to the HOMFLY polynomial.
Torsion volume forms and twisted Alexander functions on character varieties of knots
Takahiro Kitayama
(University of Tokyo)
ABSTRACT: Using non-acyclic Reidemeister torsion, we can canonically construct a complex volume form on each component of the lowest dimension of the $SL_2(\mathbb{C})$-character variety of a link group. This volume form enjoys a certain compatibility with the following natural transformations on the variety.
Two of them are involutions which come from the algebraic structure of $SL_2(\mathbb{C})$ and the other is the action by the outer automorphism group of the link group. Moreover, in the case of knots these results deduce a kind of symmetry of the $SU_2$-twisted Alexander functions which are globally described via the volume form.
A new braid index estimation using Dehornoy floor
Tetsuya Ito
(University of Tokyo)
ABSTRACT: We provide a new lower bound of the braid index of a link by using the Dehornoy floor, which is a non-negative integer defined by the Dehornoy ordering of the braid groups.
Maximal decomposition of the Turaev-Viro HQFT
Jerome Petit
(Tokyo Institute of Technology)
ABSTRACT: In a previous work, I have defined the Turaev-Viro HQFT using a group associated to a spherical category. This group is called the graduator of the category and defined a graduation on the category. I will extend the construction of the Turaev-Viro HQFT for every graduation on a spherical category. Furthermore I will show that the HQFT obtained is induced by the HQFT obtained from the graduator. To obtain this result I will define an homotopical invariant for every graduation and I will compare this invariant to the homotopical Turaev-Viro invariant which is defined for the graduator.
Pseudo diagrams of knots, links and spatial graphs and its related topics
Ryo Hanaki
(Waseda University)
ABSTRACT: We introduce a notion of pseudo diagrams and its study. We talk about related topics.
Quandle and hyperbolic volume
Ayumu Inoue
(Tokyo Institute of Technology)
ABSTRACT: We show that hyperbolic volume can be viewed as a quandle cocycle. It gives us a criterion for determining invertibility and positive/negative amphicheirality of hyperbolic knots.
Gröbner fan and its application
Tomohito Morita
(Tokyo Institute of Technology)
ABSTRACT: Gröbner bases theory has many applications for commutative algebra and algebraic geometry. In this talk, I will explain a relation between Gröbner bases theory and algebraic geometry through toric geometry.
An introduction to the volume conjecture
Hitoshi Murakami
(Tokyo Institute of Technology)
ABSTRACT: The volume conjecture would relate the colored Jones polynomial of a knot to the volume of its complement. I will give an introduction to the conjecture, starting with the definition of the colored Jones polynomial.
Introduction to nanowords
Andrew Gibson
(Tokyo Institute of Technology)
ABSTRACT: Nanowords were defined by Turaev and can be viewed as a generalization of virtual knots. In this talk I will give an introduction to the theory of nanowords using virtual knots as an example.
Introduction to quantum invariants
Jerome Petit
(Tokyo Institute of Technology)
ABSTRACT: In the early nineties, Jones introduced a new class of knot invariants, called : quantum invariants. Usually quantum invariants are obtained from an algebraic data (quantum groups, modular categories ...) and a combinatorial description of the manifolds or the links. In this talk, I will give several constructions of quantum invariants and I will talk about some applications of the quantum invariants, such as TQFT, categorification.
An introduction to Khovanov homology
Yuanyuan Bao
(Tokyo Institute of Technology)
ABSTRACT: In this talk, I will review the construction of Khovanov homology, which is due to Mikhail Khovanov.
The talk consists of three parts:
1. constructing Jones polynomial via the Kauffman bracket.
2. categorifing the Jones polynomial by using the method in Viro's paper, by which we get the Khovanov homology.
3. calculating the Khovanov homology of trefoil knot as an example.
Ptolemy Groupoid and Universal invariants
Jean-Baptiste Meilhan
(Fourier Institute, University of Grenoble)
ABSTRACT: We define an invariant of all 3-manifolds obtained by surgery in the cylinder S\times [0.1], where S is a (connected) surface with nonempty boundary. This invariant depends only on the choice of a fatgraph for S, which is essentially the dual to a triangulation of S. Also, this invariant is universal among finite type invariants of homology cylinders.
The Ptolemy groupoid of S provides a presentation of the mapping class group of S in terms of sequences of elementary moves on fatgraphs. We study the action of the Ptolemy groupoid of S on our universal invariant, and use this to give a TQFT-flavored reconstruction of the LMO invariant of an integral homology sphere given by a Heegaard splitting.
This is a joint work with J.E. Andersen, A.J. Bene and R.C. Penner.
Tunnel complexes for 3-manifolds
Yuya Koda
(Tokyo Institute of Technology)
ABSTRACT: For each closed 3-manifold M and a natural number t, we define a simplicial complex T_t(M), the t-tunnel complex, whose vertices are knots of tunnel number less than or equal to t. We then show that the complex T_1(M) is connected for M the 3-sphere or a lens space. Using this complex, we define a complexity tc(K) for tunnel number one knots K, the tunnel complexity. We show that tc(K) is less than or equal to 1 if and only if K is a (1,1)-knot for K embedded in the 3-sphere. We will mention about the relation between the above complex and Cho-McCullough's complex in the case of S^3.
三次元多様体上の葉層構造の研究と、女性研究者に必要だと思うこと
村井 紘子
（東京電機大学）
講演者からのコメント： 私は三次元多様体上の余次元1の葉層構造について研究しています。 本講演では研究の歴史的背景から、具体例を交えながら研究結果をご紹介し、その後、女性研究者になるために、そして女性が研究を続けていくために私が大切だと思い心がけていることを、質疑応答を交えながらお話します。 会場の皆さんと意見交換をしたいと思っていますので是非ご参加ください。
本セミナーは、文部科学省 科学技術振興調整費「理工系女性研究者プロモーションプログラム」により東京工業大学 男女共同参画推進センターが共同開催するものです。
Refined Kirby calculus for some homology lens spaces
Kenichi Fujiwara
(Tokyo Institute of Technology)
ABSTRACT:
Every connected oriented closed three-manifold is obtained
by Dehn surgery on an integral framed link in the three-sphere.
If two framed links present the same manifold,
then they are related by a sequence of Kirby moves.
It is known that
every integral homology three-spheres is presented by a link
with framings +/- 1 and with linking numbers 0.
Restricting to such framed links,
K. Habiro arranged Kirby moves into a set of moves so that
it preserves framings and linking numbers,
and he proved that the set of his moves
suffices to relate all framed links with the same resulting homology sphere.
We aim to generalize Habiro's result to include some homology lens spaces.
A-polynomial and the volume conjecture
Hitoshi Murakami
(Tokyo Institute of Technology)
ABSTRACT: I will talk about a relation of the A-polynomial to the volume conjecture.
Knot Floer homology of (1,1)-knots and recent topics
Hiroshi Goda
(Tokyo University of Agriculture & Technology)
ABSTRACT: We introduce the notion of a (g,b)-knot, and demonstrate a calculation of Knot Floer homology of (1,1)-knots. Further, some results of Sutured Floer homology are presented.
Colored Turaev-Viro invariants of twist knots
Yuya Koda
(Tokyo Institute of Technology)
ABSTRACT: In 2007, Barrett, Garcia-Islas and Martins defined a new series of invariants, colored Turaev-Viro invariants, of a pair (M, L), where M is a closed oriented 3-manifolds and L is an oriented link embedded in M. These invariants are defined as state-sums on a special polyhedron, restricting only to states such that certain regions have a certain pre-fixed color.
In this talk, we briefly review the definition of these invariants. Then we construct special polyhedrons for twist knots using (1,1)-decomposition of them, and we provide a formula for colored Turaev-Viro invariants of twist knots using these spines.
Homotopy of Gauss words
Andrew Gibson
(Tokyo Institute of Technology)
ABSTRACT: Homotopy is an equivalence relation on Gauss words defined using combinatorial moves. We define a homotopy invariant of Gauss words and use it to show that there are Gauss words which are not homotopic to the empty Gauss word, disproving a conjecture by Turaev. In fact, we show that the number of equivalence classes of Gauss words under homotopy is infinite.
A homotopy invariant of Gauss phrases
Andrew Gibson
(Tokyo Institute of Technology)
ABSTRACT: A Gauss phrase is a tuple (ordered list) of sequences of letters such that the concatenation of the sequences gives a Gauss word. Two Gauss phrases are homotopic if they are related by a finite sequence of certain combinatorial moves. In this talk we describe a homotopy invariant of Gauss phrases and determine the values that it can attain.
Dichromatic invariants of smooth 4-manifolds
Jerome Petit
(Tokyo Institute of Technology)
ABSTRACT: L. Crane, L.H. Kauffman and D.N. Yetter have defined an invariant of smooth and closed 4-manifolds using Cesar De Sa presentation of 4-manifolds and 4-conformed categories. For modular categories, their construction gives an invariant which is the signature of the 4 manifold. Modifying their construction and using pair of premodular categories we obtain three invariants of smooth and closed 4-manifold, called dichromatic invariants. In some cases we recover the invariant defined by Crane, Kauffman and Yetter. Furthermore we show that their invariant is always the signature of the 4-manifold. Furthermore for simply connected manifold these invariants are the signature and the Euler caracteristic.
No seminar
Categorification of the colored Jones polynomial
Jerome Petit
(Tokyo Institute of Technology)
ABSTRACT: The colored Jones Polynomial of links has two natural normalizations : one in which the n-colored unknot evalutes to [n+1], the quantum dimension of the (n+1)-irreducible representation of quantum sl2) and the other in which it evaluates to 1. For each normalization there is a bigraded cohomology theory of links with the colored Jones polynomial as the Euler characteristic. This talk is based on an article by M. Khovanov, "Categorifications of the colored Jones polynomial" , Journal of Knot Theory and its Ramifications, Vol 14, no 1, p 111-130.
On the braid index of torus links
Yuanyuan Bao
(Tokyo Institute of Technology)
ABSTRACT: In this talk, we will review some classical results on the estimation of braid index in knot theory. Especially some known properties of positive braids are given here, with which we explain in an intuitive way the well-known result: braid(L)=min(m,n), for the (m,n)-torus link.
LeBrun-Mason type twistor correspondence for Einstein-Weyl structures
Fuminori Nakata
(Tokyo Institute of Technology)
ABSTRACT: In this talk, I will introduce some recent progress on LeBrun-Mason type twistor correspondences, in particular, twistor theory for Einstein-Weyl 3-manifolds.
Using the method of LeBrun-Mason theory, we can construct infinitely many Einstein-Weyl structures on $S^2 \times R$ of signature (1,2) which is sufficiently close to the model case of constant curvature, and whose space-like geodesics are all closed. Such structures are obtained from small perturbations of the diagonal of $CP^1\times CP^1$.
On computing distances in the pants complex
Kenneth J. Shackleton
(Tokyo Institute of Technology)
ABSTRACT: The pants complex is an accurate combinatorial model for the Weil-Petersson metric (WP) on Teichmueller space (Brock). One hopes that many of the geometric properties of WP are accurately replicated in the pants complex, and this is the source of many open questions. We compare these in general, and then focus on the 5-holed sphere and the 2-holed torus, the first non-trivial surfaces. We arrive at an algorithm for computing distances in the (1-skeleton of the) pants complex of either surface.
Finite-type invariants for curves on surfaces
Noboru Ito
(Waseda University)
ABSTRACT: This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a higher order generalization of Arnold's invariants which are first order invariants for plane immersed curves. The invariants in this theory are developed using word theory proposed by Turaev.
Genus invariants of 3-manifolds associated to homology cobordisms of surfaces
Takuya Sakasai
(Tokyo Institute of Technology)
ABSTRACT: A homology cylinder is a homology cobordism between surfaces with markings of its boundary. From each homology cylinder, we can obtain a closed 3-manifold by using an operation called a "closing". It can be regarded as a homological analogue of an open book decomposition of a closed 3-manifold. We discuss three kinds of genus invariants (open book genus, homology cylinder genus and handle number) of 3-manifolds associated to homology cylinders and their closings. The contents of this talk includes a joint work with Hiroshi Goda.
Moduli of Bridgeland stable objects on the projective plane
Ryo Okawa
(Tokyo Institute of Technology)
ABSTRACT: For a projective surface X, we study the Bridgeland stability conditions on X, and we see that for some stability conditions on D(X) the moduli spaces of stable objects in D(X) coincide with the moduli spaces of stable coherent sheaves. In particular, we construct the moduli scheme of stable coherent sheaves on the projective plane as the moduli scheme of semistable quiver representations and give the birational transformation from the moduli scheme of stable coherent sheaves to another moduli space of Kronecker quiver representations.
No seminar
A natural ch-diagram of the n-twist spun trefoil knot
Ayumu Inoue
(Tokyo Institute of Technology)
ABSTRACT: In 1962, R. H. Fox introduced a series of motion pictures of $2$-knots, called the Fox's $2$-sphere. T. Kanenobu showed that they are $2$-twist spun knots, in 1983. On the other hand, the speaker has constructed another series of motion pictures that are $n$-twist spun trefoil knots ($n \geq 1$). In this talk, he explains how to get the series of motion pictures.
Turaev-Viro TQFT splitting
Jerome Petit
(Tokyo Institute of Technology)
ABSTRACT: The Turaev-Viro invariant is a 3-manifolds invariant. It is obtained in this way :
In this talk, we will give a decomposition of the Turaev-Viro TQFT. More precisely, we decompose it into blocks. These blocks are given by a group associated to the spherical category which was used to construct the Turaev-Viro invariant. We will show that these blocks define a HQFT (roughly speaking a TQFT with an "homotopical data"). This HQFT is obtained from an homotopical invariant, which is the homotopical version of the Turaev-Viro invariant. Moreover this invariant can be used to obtain the homological Turaev-Viro invariant defined by Yetter.
Hopf link stabilization in refined Kirby calculus
Kenichi Fujiwara
(Tokyo Institute of Technology)
ABSTRACT: Every integral homology sphere is presented by a framed link without linking numbers and with framing +/- 1. Habiro arranged Kirby calculus so that it preserves linking numbers and framings. Moreover, he showed that any two framed links without linking numbers present homeomorphic integral homology spheres if and only if they are related by a sequence of his moves.
Recently, the speaker obtained a generating set of an orthogonal group. In this talk, we describe a strategy to extend Habiro's result and how to apply the generating set.
No seminar
The colored Jones polynomials and representations of a knot
Hitoshi Murakami
(Tokyo Institute of Technology)
ABSTRACT: We will study the asymptotic behaviors of the colored Jones polynomials of a knot and their relations to representations of the fundamental group of its complement at the complex two-dimensional special linear group.
Some simple homotopy invariants of nanophrases
Andrew Gibson
(Tokyo Institute of Technology)
ABSTRACT: A nanophrase is an abstract generalization of a virtual link diagram, defined by Turaev. By analogy to the virtual Reidemeister moves, we can define equivalence relations on nanophrases which describe different kinds of homotopy. One such homotopy gives equivalence classes which are in bijective correspondence to virtual links. In this talk I will review the definition of nanophrases and their homotopies. I will then go on to describe some simple invariants of nanophrases under homotopy.
No seminar
A very simple introduction to the Brownian motion and the Ito formula
Yuzuru Inahama
(Tokyo Institute of Technology)
ABSTRACT: pdf file
Uniqueness of solutions to the Cauchy problem for the heat equation
Minoru Murata
(Tokyo Institute of Technology)
ABSTRACT: This talk is a brief survey on uniqueness of solutions to the Cauchy problem for the heat equation on a non-compact Riemannian manifold. We give a uniqueness theorem for solutions with some growth condition at infinity, a uniqueness theorem and a non-uniqueness theorem for nonnegative solutions. Finally, we shall present an integral representation theorem for nonnegative solutions of the heat equation.
Winter holiday
No seminar
Favorite site of simple random walk and Brownian motion
Naoto Takahashi
(Tokyo Institute of Technology)
ABSTRACT: We survey asymptotic properties of the most visited site (i.e. favorite site) for simple random walk or Brownian motion. We also introduce open problems about favorite site.
No seminar
Teichmüller spaces and holomorphic families of Riemann surfaces
Hiroshige Shiga
(Tokyo Institute of Technology)
ABSTRACT: This talk is a brief survey of our works on the theory of Teichmüller spaces of Riemann surfaces and its applications. Particularly, we will focus the argument on an interaction between analysis and geometry.
The double Riemann zeta function
Hirotaka Akatsuka
(Tokyo Institute of Technology)
ABSTRACT: The absolute tensor product was defined by Kurokawa (1992) as an attempt to investigate zeros of zeta functions. The absolute tensor product constructs a meromorphic function (a multiple zeta function) from some ordinary zeta functions by using their zeros and poles. In this talk, starting from a review on the Riemann zeta function from a viewpoint of prime numbers and its zeros, we give the Euler product expression for the double Riemann zeta function constructed from the absolute tensor product.
A homological characterization of surface coverings
Masaharu Tanabe
(Tokyo Institute of Technology)
ABSTRACT: We characterize - in terms of matrices - the abstractly given homomorphisms of the first homology groups of closed oriented surfaces which can be induced by coverings of prime degree. We also classify the induced homomorphisms in these cases.
No seminar
On Picard numbers of Fano varieties
Toru Tsukioka
(Tokyo Institute of Technology)
ABSTRACT: The projective varieties with ample anti-canonical bundles are called Fano varieties (these can be considered as generalization of projective spaces). They are of great interest in algebraic geometry and smooth ones are completely classified up to (complex) dimension 3. It is known that smooth Fano varieties (defined over the complex number field) have only a finite number of deformation types in each dimension. In particular, the Picard number of a smooth Fano variety (which coincides with the second Betti number) is bounded by a function of the dimension. However, the explicite bounds are unknown in dimensions greater than or equal to 4. In this talk, we discuss the bound of Picard numbers for certain smooth Fano varieties in higher dimensions.
Similarity transformations, Quantization and Classical Mechanics
Atsushi Inoue
(Titech)
ABSTRACT: pdf file
Seifert surgeries and Montesinos trick
Arnaud Deruelle
(Titech)
joint with M. Eudave-Munoz, K. Miyazaki and K. Motegi
ABSTRACT: I am interested in Dehn surgery on knots in S3, and in particular in Seifert fiber spaces obtained by such a construction. I will (re-)introduce first a Seifert Surgery Network from a previous joint work with K. Miyazaki and K. Motegi. Then I will discuss a family introduced by M. Eudave-Munoz, and locate them in the Network. These knots are defined by using the so-called Montesinos trick so I will recall this construction.
An application of arithmetic group theory
Kenichi Fujiwara
(Titech)
ABSTRACT: We want to find a finite generating set of a discrete Lie subgroup. For example, we provide the well-known pair of generating matrices (up to signature) of the two-dimensional special linear group with integral coefficients. To do this, we shall compute integral points of a Siegel set.
A unified theorem describing
Hopf and pitchfork bifurcation and
applications to computer assisted proofs
Tadashi Kawanago
(Department of Mathematics,
Tokyo Institute of Technology)
ABSTRACT: We will present a unified bifurcation theorem describing Hopf and pitchfork bifurcation. We can apply it to the computer assisted existence proofs of bifurcation points for given dynamical systems described by partial differential equations.
On a ball in a metric space of knots by delta moves
Sumiko Horiuchi
(Tokyo Woman's Christian University)
ABSTRACT: We consider the metric space of all knots on which the distance is defined by delta moves. We show that for any two knots $K_1$ and $K_2$ with delta distance $k$ and for any natural numbers $\ell$ and $m$ with $\ell+m=k$, the intersection of the ball of radius $\ell$ centered at $K_1$ and the ball of radius $m$ centered at $K_2$ contains infinitely many knots. We also consider the problem whether or not the center of a given ball is unique.
A partial order on the set of prime knots
and representations of knot groups II
Masaaki Suzuki
(Akita University)
ABSTRACT: A relation between two prime knots is defined if there exists a surjective homomorphism between knot groups. It is known that this relation is a partial order on the set of prime knots. In this talk, we determine this partial order in Rolfsen's knot table. Furhermore, we study SL(2,Z/pZ)-representations of knot groups. They play an important role in proving the non-existence of a surjective homomorphism.
Invariants via word for curves
Noboru Ito
(Waseda University)
ABSTRACT: We construct the infinite sequence of invariants for curves in surfaces by using word theory that V. Turaev introduced. For plane closed curves, we add some extra terms, e.g. the rotation number. From these modified invariants, we get the Arnold's basic invariants and some other invariants. We also express how these invariants classify plane curves.
NO seminar
Invariants of knots with exceptional Dehn surgeries
Yukihiro Tsutsumi
(Sophia University)
ABSTRACT:
A Dehn surgery on a knot K is exceptional if K is hyperbolic and
the manifold obtained by the surgery is not hyperbolic.
My talk consists of the following three parts:
(1) Summary of recent works on exceptional surgeries,
(2) Structures of manifolds obtained by toroidal surgeries,
(3) How to compute Casson invariants of cyclic covering spaces branched along knots with Lens
surgeries.
A combinatorial formula on convex polytopes
Chikara Nakayama
(Tokyo Institute of Technology)
ABSTRACT: As an introduction to some algebraic theories, we explain an arithmetic proof of the Dehn-Sommerville equalities for simplicial convex polytopes, based on the theory of toric varieties and the weights of $l$-adic cohomologies.
Absolutely continuous invariant measures for expansive diffeomorphisms of the 2-torus
Naoya Sumi
(Department of Mathematics, Tokyo Institute of Technology)
joint work with M. Hirayama (Hiroshima University)
ABSTRACT: The aim of this talk is to obtain an equivalent criterion for certain expansive diffeomorphisms of ${\mathbb T}^2$ to admit an invariant Borel probability measure that is absolutely continuous with respect to the Riemannian volume. Our result is closely related to the well known Liv\v{s}ic-Sinai theorem for Anosov diffeomorphisms.
NO seminar
Seifert surgeries and Montesinos trick
Arnaud Deruelle
(Titech)
joint with M. Eudave-Munoz, K. Miyazaki and K. Motegi
ABSTRACT: I am interested in Dehn surgery on knots in S3, and in particular in Seifert fiber spaces obtained by such a construction. I will (re-)introduce first a Seifert Surgery Network from a previous joint work with K. Miyazaki and K. Motegi. Then I will discuss a family introduced by M. Eudave-Munoz, and locate them in the Network. These knots are defined by using the so-called Montesinos trick so I will recall this construction.
Coverings of Virtual Strings
Andrew Gibson
(Titech)
ABSTRACT: In his paper 'Virtual Strings and Their Cobordisms' Turaev defines coverings of virtual strings. A covering of a virtual string is a homotopy invariant of the string and is itself a virtual string. Thus we can consider coverings as maps from the set of virtual strings to itself. In this talk I will explain Turaev's definition and show that the set of fixed points in each covering map is infinite.
NO seminar
An introduction to the Kontsevich integral for curious cats
Hitoshi Murakami (Titech)
ABSTRACT: I will introduce the Kontsevich integral of a knot.
Scissors Congruences and group homology of Lie groups
Yuichi Kabaya (Titech)
ABSTRACT: Two polyhedra in Euclidean (, spherical or hyperbolic) space are scissors congruent if they can be divided into smaller pieces that are congruent. The study of scissors congruence is related to the group homology of Lie group, for example Euclidean case the Lie group is O(n). I will talk about the relationship between scissors congruence and group homology.
No seminar
Knots and noncompact 4-manifolds
Toshifumi Tanaka (Osaka City University, COE Fellow)
ABSTRACT:Does every noncompact 4-manifold admit at least two different smooth structures? We investigate the relationship between knots and links in the 3-sphere and the diffeomorphism types of homeomorphic smooth 4-manifolds. We show that every noncompact, connected oriented smooth 4-manifold with a knot, called a z-exotic knot, admits at least two smooth structures. In fact, every noncompact, connected smooth 4-submanifold of the complex projective plane admits a z-exotic knot.
On the Alexander polynomial satisfying Ozsváth-Szabó's condition for lens sugery
Teruhisa Kadokami (Osaka City University, COE researcher)
ABSTRACT: If a knot K in S3 has a lens surgery, then the Alexander polynomial ΔK(t) of K satisfies some conditions. We study Ozsváth-Szabó's condition from knot Floer homology and Kadokami-Yamada's condition from Reidemeister torsion. By combining them, we obtained two results: (1) If ΔK(t) has the form ΔK(t)=ΔT(r, s)(tn) where T(r, s) is the (r, s)-torus knot and n is a positive integer, then we have n=1. (2) Between parameters in Ozsváth-Szabó's condition and those in Kadokami-Yamada's condition, an equation holds. It is proved by computing Conway's a2-term of K. We show some applications.
Khovanov homology and Rasmussen's s-invariants for pretzel knots
Ryouhei Suzuki (University of Tokyo)
ABSTRACT: We calculated the rational Khovanov homology of some class of pretzel knots, by using the spectral sequence constructed by P. Turner. Moreover, we determined the Rasmussen's s-invariant of almost of pretzel knots with three pretzel.
On the Reidemeister torsion associated to knots
Yoshikazu Yamaguchi (University of Tokyo)
ABSTRACT: We review the Reidemeister torsion associated to knots, sometimes called the twisted Reidemeister torsion. We will also explain examples of computation for torus knots by using Fox calculus.
NO seminar
The pants graph: Geometry and applications
Kenneth J. Shackleton (JSPS, PD)
ABSTRACT: We survey the pants graph after Hatcher-Thurston, and discuss some of its applications in modern low-dimensional topology. Related topics include theory of mapping class groups, knot theory, Teichmueller space (with Weil-Petersson metric, from the work of Brock), 3-manifolds. Time permitting, we will also try to understand some of its important subgraphs.
Visualizing the Magnus representation of braid groups
Fumikazu Nagasato (JSPS, PD)
ABSTRACT: We have already shown that ``a prototype'' of the Magnus representation can be constructed diagrammatically by using the KBSM of a handlebody. Here we note that the KBSM of a handlebody can be thought of as a quotient polynomial ring. Then introducing a method to lift the KBSM of a handlebody to the free polynomial ring diagrammatically, we lift the prototype of the Magnus representation to the genuine Magnus representation with a diagrammatic description.