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Speaker ：Takahito KURIYA (Faculty of Mathematics, Kyushu University, JSPS) Title ：Remark on a spin refinement of the perturbative invariant   （Abstract）    （PDF） Date ：January 20(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Taiji Taniguchi (Keio University) Title ：Turaev-Viro invariant for all orientable Seifert manifolds   （Abstract）    （PDF） Date ：December 16(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Yuya Koda (Department of mathematics, Keio University) Title ：Heegaard-type presentations of branched standard spines and Reidemeister-Turaev torsion   （Abstract）    （PDF） Date ：December 9(Fri.) 16：10～17：10 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Rama Mishra (JSPS Researcher, Osaka City University,                       Indian Institute of technology-Delhi, India) Title ：In search of an ideal polynomial representation of a knot-type   （Abstract）    （PDF） Date ：December 9(Fri.) 15：00～16：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Taizo KANENOBU (Department of Mathematics, Osaka City University) Title ：Skein Relation for the HOMFLYPT Polynomials of Two-Cable Links   （Abstract）    （PDF） Date ：December 2(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Yukihiro Tsutsumi (Sophia University, JSPS Research Fellow) Title ：On the equivariant Casson invariant for fibered knots   （Abstract）    （PDF） Date ：November 25(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Eiko Kin (Kyoto University Research fellow) Title ：A search of braids which have smaller dilatation than given pseudo-Anosov braids   （Abstract）    （PDF） Date ：November 18(Fri.) 13：30～14：30 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Nafaa Chbili (OCAMI, COE Research member) Title ：Symmetry in Dimension three: A quantum approach   （Abstract）    （PDF） Date ：November 4(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：KAWAMURA, Tomomi (Aoyama Gakuin University ) Title ：The Rasmussen invariants and the sharper slice-Bennequin inequality on knots   （Abstract）    （PDF） Date ：October 28(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：TANAKA, Kokoro (University of Tokyo/JSPS) Title ：Inequivalent surface-knots with the same knot quandle   （Abstract）    （PDF） Date ：October 21(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：KODAMA, Kouji (Kobe City College of Technology) Title ：Computing technic of knot theory   （Abstract）    （PDF） Date ：October 7(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：KAMADA, Naoko (Osaka City University) Title ：An algorithm to calculate Miyazawa polynomials of virtual knots   （Abstract）    （PDF） Date ：September 30(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Fengchun Lei （Harbin Institute of Techonology） Title ：On Maximal Collections of Essential Annuli in a Handlebody   （Abstract）    （PDF） Date ：September 2(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：YAMAMOTO, Minoru (Hokkaido University) Title ：The minimal number of singular set for fold maps   （Abstract）    （PDF） Date ：July 15(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：YASHIRO, Tsukasa(JSPS fellow,Osaka City University) Title ：Cross-exchangeable cycles and 1-handles for surface diagrams   （Abstract）    （PDF） Date ：July 8(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Tom Fleming（University of California, San Diego (UCSD)） Title ：Chirality of Alternating Knots in S x I   （Abstract）    （PDF） Date ：July 1(Fri.) 17：00～18：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：TANAKA, Toshifumi (University of Tokyo) Title ：Ribbon 2-knots assoiciated with symmetric unions   （Abstract）    （PDF） Date ：July 1(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：TAYAMA, Ikuo (OCAMI) Title ：Enumerating the exteriors of prime links by a canonical order   （Abstract）    （PDF） Date ：June 24(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：SUZUKI, Masaaki(University of Tokyo) Title ：Twisted Alexander invariant and a partial order in the knot table   （Abstract）    （PDF） Date ：June 10(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：ICHIHARA, Kazuhiro(Osaka Sangyo University) Title ：Alexander polynomials of doubly primitive knots     (joint work with Toshio Saito (Osaka University)      and Masakazu Teragaito (Hiroshima University))   （Abstract）    （PDF） Date ：June 3(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：SATOH, Shin（Chiba University） Title ：Disk presentations of surface-knots and -links   （Abstract）    （PDF） Date ：May 20(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：NAKANISHI, Kiyotaka（Kyushu University） Title ：Sliding relation in the Kauffman bracket skein module of a 2-bridge knot exterior   （Abstract）    （PDF） Date ：May 13(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：ISHII, Atsushi（Osaka University） Title ：Skein relations for the generalized Alexander polynomial for virtual links and closed virtual 2-braids   （Abstract）    （PDF） Date ：May 6(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top

Speaker：	Takahito KURIYA (Faculty of Mathematics, Kyushu University, JSPS)
Title：	Remark on a spin refinement of the perturbative invariant

In this talk, we will give a brief review of a spin refinement of the perturbative invariant defined by A.Beliakova, C.Blanchet and T.Le, and we will try to construct a invariant of spin 3-manifolds.
When the time allows, we talk on some related topics.

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Speaker：	Taiji Taniguchi (Keio University)
Title：	Turaev-Viro invariant for all orientable Seifert manifolds

Turaev-Viro invariant is a topological invariant for closed 3-manifolds. In this talk, we introduce the definition of TV-invariant by using special spines of closed 3-manifolds, and give a formula of TV-invariant for all orientable Seifert manifolds. Our formula is based on a new construction of special spines of all orientable Seifert manifolds and the "gluing lemma" of topological quantum field theory. By using this formula, we get sufficient conditions that TV-invariant of Seifert manifolds coincides and make a computer program calculating TV-invariant of orientable Seifert manifolds. The method of constructing our formula is applicable to the "state sum" type invariant, for example Turaev-Viro-Ocneanu invariant or Dijkgraaf-Witten invariant.

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Speaker：	Yuya Koda (Department of mathematics, Keio University)
Title：	Heegaard-type presentations of branched standard spines and Reidemeister-Turaev torsion

Reidemeister-Turaev torsion is an invariant of a 3-manifold $M$ equipped with a $\mathrm{Spin}^c$ structure, one representation of which is a homology class of a non-singular vector field on $M$. In this talk, we introduce a way to represent a branched standard spine, which can be regarded as a combinatorial presentation of a $\mathrm{Spin}^c$ structure on a 3-manifold, as a Heegaard diagrams with "mark", and explain an accessible way to compute the invariant using this presentation. We also explain some interesting behavior of this invariant by using examples, and relation to Seiberg-Witten invariant.

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Speaker：	Rama Mishra (JSPS Researcher, Osaka City University,                       Indian Institute of technology-Delhi, India)
Title：	In search of an ideal polynomial representation of a knot-type

One of the central themes of {\it geometric knot theory} is to find an ideal configuration of knot-types in a specified category. Here we consider the space $\mathbb{P}$ of all polynomial knots. Following Vassiliev's notation $\mathbb{P}=K3\setminus \Sigma,$ where $K3$ is the set of all maps $\phi:\mathbb{R}\longrightarrow\mathbb{R}^3$ defined by $\phi(t)=(x(t),y(t),z(t))$ where $x(t)=t^d+a_1t^{d-1} +\cdots+a_{d-1}t$, $y(t)= t^d+b_1t^{d-1}+\cdots+b_{d-1}t,$ and $z(t) =t^d+c_1t^{d-1}+\cdots+c_{d-1}t)$ and $\Sigma$ is the descriminent space of $K3.$ The topology of the space $\mathbb{P}$ is still under investigation. It is proved that any $C1$ knot in $S3$ is isotopy equivalent to the closure of the image of such a map for some degree $d$. Also the path components of $\mathbb{P}$ correspond to a knot-type. It can be easily proved that nontrivial knots cannot be realized by such maps of degree less than 5. As, polynomial represenatation for a given knot-type is not unique, the question of choosing an ideal poynomial representation makes sense. We have made an effort to define an energy function on the space $\mathbb{P}$ and based on this function we call a polynomial representation of a given knot-type with minimum energy to be the ideal one. We shall discuss if such an ideal representation for a given knot-type exists?

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Speaker：	Taizo KANENOBU (Department of Mathematics, Osaka City University)
Title：	Skein Relation for the HOMFLYPT Polynomials of Two-Cable Links

We give a skein relation for the HOMFLYPT polynomials of 2-cable links. In [A. Ishii and T. Kanenobu, Different links with the same Links-Gould invariant, Osaka J. Math. 42 (2005) 273--290], we construct examples of arbitrarily many 2-bridge knots sharing the same HOMFLYPT, Kauffman, and Links-Gould polynomials, and arbitrarily many 2-bridge links sharing the same HOMFLYPT, Kauffman, Links-Gould, and 2-variable Alexander polynomials. Using the skein relation, we show their 2-cables also share the same HOMFLYPT polynomials.

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Let $K$ be a fibered knot in $S3$ with ${\rm vol}(S3-K)=0$. Then $K$ is a fibered knot such that the monodromy is decomposed into periodic maps and the JSJ-family of the complement consists of Seifert fibered spaces. A theorem of O. Collin and N. Saveliev implies that for such a fibered knot $K$, the Casson invariant $\lambda(\Sigma^r_K)$ of the $r$-fold cyclic cover $\Sigma^r_K$ of $S3$ with branch set $K$ is written in terms of the equivariant knot-signatures of $K$ which are determined by the matrix of the monodromy (when $\Sigma^r_K$ is an integral homology sphere.) This is specific to fibered knots with ${\rm vol}(S3-K) = 0$ and is due to some property of the $SU(2)$-representation of $\pi_1(\Sigma^r_K)$ under the cyclic action on $\Sigma^r_K$.

Given a fibered knot with certain properties, one can construct infinitely many fibered knots with the same Seifert form and with distinct values of $\lambda(\Sigma^r_K)$ by adding a Dehn twist to the monodromy. We study the variation of $\lambda(\Sigma^r_K)$ under this construction and ${\rm vol}(S3-K)$ for fibered knots and non-fibered knots with trivial Alexander polynomial.

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The dilatation is an invariant of pseudo-Anosov braids. Fixing strands n, the minimal of dilatation exists among pseudo-Anosov n-braids. Toward the determination of braids with minimal dilatation, we discuss how to find braids which have smaller dilatation than given pseudo-Anosov braids. A result by Los tells us that a given pseudo-Anosov braid $\beta$, any braid $\alpha$ dynamically forced by $\beta$ have always smaller dilatation than $\beta$. Such forced braid $\alpha$ can be captured by using the train track map associated to $\beta$. A problem is to know whether $\alpha$ is pseudo-Anosov or not, and how to compute the train track map associated to $\alpha$ when it is pseudo-Anosov. This is because in general, one can not predict how does the train track map of $\alpha$ look like. We give a solution of the problem when the train track map of $\beta$ contains "the star shaped rotational map".

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Let $M$ be an oriented compact three-manifold and $G$ a finite cyclic　group of prime order. The manifold $M$ is said to be symmetric if $G$ acts non trivially on $M$. Several classical methods have been used to study the symmetries of three-manifolds. In this talk, we shall explain how to use the quantum invariants to study this problem. Namely, we will show how Murasugi's results on periodic knots have been extended to quantum invariants of three-manifolds by Chbili and Gilmer in the case of the $Su(2)$ and the $Su(3)$ quantum invariants. Then, by Chen-Le to allcomplex simple Lie algebras and by Qazaqzeh to modular categories.

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Rasmussen introduced a knot invariant based on Khovanov homology theory, and showed that this invariant estimates the four-genus. We compare his result with the sharper slice-Bennequin inequality for knots. Then we obtain a similar estimate of the Rasmussen invariant to this inequality.
It includes the Bennequin inequality on the Rasmussen invariants, which is shown independently by Plamenevskaya and Shumakovitch.

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We have a knot quandle $Q(k)$ and a fundamental class $[k]$ as invariants for a classical knot $k$.
Similarly, we have a knot quandle $Q(F)$ and a fundamental class $[F]$ as invariants for a surface-knot $F$.

For classical knots, Joyce and Matveev independently proved that $Q(k)$ characterizes the classical knot $k$ up to reflected inverse,and Eisermann proved that the pair $Q(k)$ and $[k]$ characterize the classical knot $k$ completely.

We consider the following "hierarchy" for surface-knots $F$ and $F'$.
(i) There exists a quandle isomorphism $\phi :Q(F) \rightarrow Q(F')$.
(ii) There exists a quandle isomorphism $\phi :Q(F) \rightarrow Q(F')$ such that $\phi_{\ast}[F] = [F']$.
(ii)' There exists a quandle isomorphism $\phi :Q(F) \rightarrow Q(F')$ such that $\phi_{\ast}[F] = \pm [F']$.
(iii) The surface-knot $F$ is equivalent to $F'$.
(iii)' The surface-knot $F$ is equivalent to $F'$ or $-(F')^{\ast}$.

We note that (iii) $\Rightarrow$ (ii) $\Rightarrow$ (i) and (iii)' $\Rightarrow$ (ii)' $\Rightarrow$ (i) by definition.

In this talk, we illustrate the gap between (i) and (ii)', the gap between (ii)' and (iii)', and the gap between (ii) and (iii)
for surface-knots.

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"KNOT" is a computing tool on knot theory. In this talk, we will study how to implement or write topological idea as programming codes. Let us walk through source codes about skein invariants in "KNOT" program.

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In 2004, Y. Miyazawa discovered a method to define polynomial invariants for a virtual knot, which are generalization of Kauffman's f-polynomial.
They enable us to distinguish Kishino's knot from a trivial knot. This August we announced a table of virtual knots with four real crossings classified by use of Miyazawa polynomials, JKSS invariants and 2-cabled Jones polynomials. In order to get the table and calculate invariants, we made a computer program. In this talk, we introduce an algorithm to calculate two kinds of Miyazawa polynomials from a Gauss chord diagram of a virtual knot.

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It is known that, for a maximal collection $\cal A$ of pairwise disjoint non-parallel essential annuli in a handlebody of genus 2, $1\leq|{\cal A}|\leq 3$. We show that for a maximal collection $\cal A$ of pairwise disjoint non-parallel essential annuli in a handlebody of genus $n$ $(\geq 3)$, $|{\cal A}|\leq 4n-5$, and the bound is best possible.

This is a joint work with Jingyan Tang.

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In 1970's, Eliashberg showed that every smooth map between oriented surfaces is homotopic to a fold map. A fold map is a smooth map which has only fold singularity ($A_1$-type singularity). It is known that for a fold map between surfaces, singular set is a one-dimensional submanifold in the source surface. In this talk, we will determine the minimal number of components of singular set for fold maps when we fix the mapping degree and the genuses of source and target oriented closed surfaces.

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A surface-knot is an embedded closed oriented surface in 4-space.
A surface diagram is the image of a projection of a surface-knot into 3-space with crossing information.
For every surface-knot, we can attach some 1-handles to obtain a trivial surface. The minimal number of such 1-handles is called the unknotting number of the surface-knot.
Some surface diagrams have special double curves, in which we can change the crossing information to obtain a trivial surface.
In this talk we will discuss about a relation between the number of such special double curves, which may have multiple points and the number of 1-handles needed to obtain a trivial surface.

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Many alternating knots, such as the figure-eight knot, are isotopic to their mirror image in $S^{3}$. Detecting amphichiral knots in $S^{3}$ can be difficult.  The situation for alternating knots embedded in a surface cross an interval is quite different. There is a well defined projection in a such a space, and we define the mirror image of a knot to be the reflection across the projection surface.  We can then show if an alternating knot K is non-trivially embedded in S x I, where the genus of S is greater than zero, then K is not isotpic to its mirror image. In other words, such a knot is chiral.

We prove this result by studying the span of a generalized Kauffman polynomial of the knot, as well as the polynomial of lifts of the knot in certain covering spaces of S.  The condition of "nontrivially embedded" is necessary to avoid the obvious counterexample induced by embedding a small $S^{3}$ into S x I.

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A symmetric union was introduced by Kinoshita and Terasaka in the 1950's, which is a generalization of the connected sum operation for a knot and its mirror image. It has been generalized by Lamm recently. Every symmetric union is a ribbon knot and it has been shown that a ribbon knot with crossing number less than or equal to ten is a symmetric union. We ask if every ribbon 2-knot has a symmetric union as an equatorial cross section (i.e. every ribbon 2-knot is assoiciated with symmetric union) because every 2-knot has a ribbon knot as an equatorial cross section. In this talk, we prove that every ribbon 2-knot of 1-fusion is assoiciated with a symmetric union for the unknot or a 2-bridge knot. We also introduce a banded symmetric union to study a ribbon 2-knot and we generalize a result of Kanenobu concerning some family of ribbon 2-knots.

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This is a joint work with Akio Kawauchi. He gave a well-order to the set of links, which induces the well-orders into the set of link exteriors and the set of 3-manifolds. The length of a link is the minimum string number when we deform the link into a closed braid. The length of a link exterior and that of a 3-manifold is defined by using the length of a link. We enumerated the prime links with up to length 10. Our goal of this study is to enumerate    3-manifolds with up to length 10. In this talk, we enumerate the exteriors of prime links with up to length 9, which is in a step of the goal..

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Twisted Alexander invariant is defined for a finitely presentable group $G$ and a representation of $G$ and a surjective homomorphism of $G$ to a free abelian group. In this talk, we introduce some examples and some properties of the twisted Alexander invariant. Moreover, as an application, we consider a partial order on the set of prime knots.
Let $K$ be a knot and $G(K)$ the knot group. For two prime knots $K,K'$, we write $K \geq K'$, if there exists a surjective homomorphism from $G(K)$ to $G(K')$. We determine this partial order ``$\geq$'' on the set of prime knots in the Rolfsen's table.

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It is an interesting open problem to determine the knots in the 3-sphere admitting a lens space surgery. Known such examples are only knots called `doubly primitive', and it is conjectured that they are all. Recently, the Alexander polynomials of such knots have been studied by Ozsvath and Szabo, Kadokami, and Yamada. In this talk, we give a formula for Alexander polynomials of doubly primitive knots. This also gives a practical algorithm to determine the genus of any doubly primitive knot.

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We introduce a new way of presenting surface-knots and -links in $S^4$, called disk presentation. This enables us to define the disk index $¥Delta(F)$ of a surface-knot or -link $F$. We prove that $¥Delta(F)=2$ if and only if $F$ is a trivial $S^2$-knot, and $¥Delta(F)=3$ if and only if $F$ is a trivial non-orientable surface-knot with $|e(F)| ¥leq  3-¥chi(F)$, where $e(F)$ is the normal Euler number of $F$ and $¥chi(F)$ is the Euler characteristic of $F$.

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In this talk, we give an expicit formula for an underlying relation in the Kauffman bracket skein module of the exterior of the $2$-bridge knot $S(2pq+1, 2q)$, which is called the ``sliding relation" in this talk. This relation comes from a sliding of the trivial knot in a genus $2$ handlebody along a simple closed curve (attaching slope) along which a $2$-handle is attached to obtain the exterior of a $2$-bridge knot, and we show that the relation is essential. We also give an application of the formula to $SL(2, \mathbb{C})$ character variety of the fundamental group of the $2$-bridge knot complement. Finally we research the commensurability of such $2$-bridge knot exteriors.

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We show how to find a skein relation for the generalized Alexander polynomial for virtual links.
   
A quantum invariant for classical links is defined by associating a vector space to a strand. A linear relation among linear maps associated to oriented classical tangles is a skein relation which is helpful for evaluating the invariant. Since a linear map associated to an oriented classical tangle is an intertwiner which is equivariant with respect to the action on $V \otimes V$, we may find a skein relation among any $n$ oriented classical tangles if $n$ is greater than the dimension of the space of intertwiners.
   
On the other hand, a quantum invariant for virtual links is in the different situation from above. A linear map associated to an oriented virtual tangle is not an intertwiner any more. So we might appreciate an alternative way to find a skein relation for a quantum invariant for virtual links. We focus on the generalized Alexander polynomial for virtual links. We introduce a finite dimensional vector space which includes all linear maps associated to oriented virtual tangles, and gave the dimension of the vector space.
   
Furthermore, by using this vector space, we find a new skein relation. As an application of the relation, we give a formula for the invariant for a closed virtual $2$-braid

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