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 Friday Seminar on Knot Theory（2008） (2007) (2009)
Organizer in 2008 : TANAKA, Toshifumi
Speaker ：Jean-Baptiste Meilhan (Grenoble University)
Title ：On finite type (concordance) invariants of string links
(joint work with Akira Yasuhara and Tomoaki Sase)
（Abstract） （PDF）
Date ：February 20 (Fri.) 16：10～17：10
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Teruhisa Kadokami (Dalian University of Technology)
Title ：An integral invariant from the knot group
(joint work with Zhiqing Yang (Dalian University of Technology))
（Abstract） （PDF）
Date ：February 20 (Fri.) 15：00～16：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Ryuji Higa (Kobe University)
Title ：Tangle sum of alternating tangles yielding a splittable link
（Abstract） （PDF）
Date ：February 6 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Hiromasa Moriuchi (OCAMI, Osaka City University)
Title ：An enumeration of non-prime theta-curves and handcuff graphs
（Abstract） （PDF）
Date ：January 30(Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Koya Shimokawa (Saitama University)
Title ：Site-specific recombination of DNA and Dehn surgery
（Abstract） （PDF）
Date ：January 23 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Andrei Pajitnov (Univ. de Nantes)
Title ：On the Morse-Novikov number and the tunnel number of knots
（Abstract） （PDF）
Date ：December 19(Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Ayumu Inoue (Tokyo Institute of Technology)
Title ：Quandle and hyperbolic volume
（Abstract） （PDF）
Date ：December 12 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Tomomi Kawamura (Graduate school of Mathematics, Nagoya University)
Title ：An estimate of the Rasmussen invariant for links
（Abstract） （PDF）
Date ：December 5 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Takuji Nakamura (Osaka Electro-Communication University)
Title ：Local moves and Periodic knots
（Abstract） （PDF）
Date ：November 28 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Shin Satoh (Kobe University)
Title ：Triviality of 2-knot with three sheets
（Abstract） （PDF）
Date ：November 21 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Taizo KANENOBU (Osaka City University)
Title ：H(2)-unknotting number of a knot
（Abstract） （PDF）
Date ：November 14 (Fri.) 15：30～16：30
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Kanako Oshiro (Hiroshima university, JSPS Research Fellow)
Title ：Estimate of minimal triple point numbers of surface-links by symmetric quandle cocycles
（Abstract） （PDF）
Date ：November 7 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Kazuhiro Ichihara (Nara University of Education)
Title ：Seifert fibered surgeries on Montesinos knots
(based on joint works with In Dae Jong and Shigeru Mizushima)
（Abstract） （PDF）
Date ：October 31 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Yoshihiro Fukumoto (Tottori University of Environmental Studies)
Title ：Bounding genera of several infinite families of Brieskorn homology 3-spheres
（Abstract） （PDF）
Date ：October 24 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Ryosuke YAMAMOTO (Osaka City University, OCAMI)
Title ：An invariant of fibered knots from the third Johnson-Morita
homomorphism of mapping class group
（Abstract） （PDF）
Date ：October 17 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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(Department of Applied Mathematics, Dalian University of Technology)
Title ：Twisted Alexander invariants of groups
（Abstract） （PDF）
Date ：October 10 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Akio KAWAUCHI (Osaka City University)
Title ：The warping degree and the unknotting number of a spatial graph
（Abstract） （PDF）
Date ：October 3 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Toshifumi Tanaka, (Osaka City University Advanced Mathematical Institute)
Title ：Signed Gordian distance and Jones polynomials
（Abstract） （PDF）
Date ：July 11 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Motoo Tange,
(RIMS-Research Institute for Mathematical Sciences Kyoto University)
Title ：The alternating condition by Ozsvath and Szabo, and the wave theorem
by Homma, Ochiai, and Takahashi
（Abstract） （PDF）
Date ：June 27 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Kouichi Yasui, (Graduate School of Science, Osaka University)
Title ：Elliptic surfaces without 1-handles
（Abstract） （PDF）
Date ：June 20 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Kengo Kishimoto, (Graduate School of Science, Osaka City University)
Title ：The dealternating number and the alternation number of a closed 3-braid
(joint work with Tetsuya Abe (Graduate School of Science, Osaka City University))
（Abstract） （PDF）
Date ：June 13 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Kokoro Tanaka, (Department of Mathematics, Tokyo Gakugei University)
Title ：A categorification of the one-variable Kamada-Miyazawa polynomial
(joint work with Atsushi ISHII (RIMS, Kyoto University))
（Abstract） （PDF）
Date ：June 6 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Daniel Moskovich (Osaka City University Advanced Mathematical Institute)
Title ：Towards surgery presentations of metabelian coloured knots and their covering links
（Abstract） （PDF）
Date ：May 30 (Fri.) 15：00～16：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：In Dae JONG, (Graduate School of Science, Osaka City University)
Title ：On a characterization of the Alexander polynomials of alternating knots of genus two
（Abstract） （PDF）
Date ：May 23 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Tetsuya Abe (Graduate School of Science, Osaka City University)
Title ：On the Turaev genus
（Abstract） （PDF）
Date ：May 16 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Tamas Kalman
(Graduate School of Mathematical Sciences, The University of Tokyo)
Title ：The problem of maximum Thurston-Bennequin number for knots
（Abstract） （PDF）
Date ：May 9 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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(Department of Applied Mathematics, Dalian University of Technology)
Title ：Numerical invariants from knot groups
(joint work with Ruifeng Qiu (Dalian University of Technology)
and Zhiqing Yang (Dalian University of Technology))
（Abstract） （PDF）
Date ：May 9 (Fri.) 15：00～16：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Ikuo Tayama (Osaka City University Advanced Mathematical Institute)
Title ：A complete classification of 3-manifolds with lengths up to 9 by a canonical order
（Abstract） （PDF）
Date ：May 2 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Kazuhiro Sakuma (Kinki University)
Title ：A question on the special values of Alexander polynomials
which detect the smooth structure on spheres
（Abstract） （PDF）
Date ：April 25(Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Masahide Iwakiri (Osaka City University Advanced Mathematical Institute)
Title ：Calculation of quandle cocycle invariants of Suzuki's $\theta_n$-curves
（Abstract） （PDF）
Date ：April 18(Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Toshifumi Tanaka (Osaka City University Advanced Mathematical Institute)
Title ：The quasipositivation number of a knot
（Abstract） （PDF）
Date ：April 11(Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
Top

 Abstracts

 Speaker： Jean-Baptiste Meilhan (Grenoble University) Title： On finite type (concordance) invariants of string links (joint work with Akira Yasuhara and Tomoaki Sase)

A $C_n$-move is a local move on links defined by Habiro and Goussarov, which can be regarded as a higher order crossing change', and which gives a complete topological characterization of the informations contained by Goussarov-Vassiliev (or finite type) knots invariants: two knots cannot be distinguished by finite type invariants of degree $<n$ if and only if they are related by a finite sequence of $C_n$-moves.
The analogous statement is known to be false for links in general, but it is conjecturally true for string links, which are certain links with boundary.
This conjecture is partly supported by the fact that Milnor invariants, which are invariants (of both links and string links) generalizing the linking number, are of finite type only for string links.
In this talk, we will classify $l$-component string links up to $C_n$-move for $n\le 5$, by explicitely giving complete sets of low degree finite type invariants. In addition to Milnor invariants, these include several new' string link invariants constructed by evaluating knot invariants on certain closure of (cabled) string links.
We also give similar results for concordance finite type invariants.

 Speaker： Teruhisa Kadokami (Dalian University of Technology) Title： An integral invariant from the knot group (joint work with Zhiqing Yang (Dalian University of Technology))

This is a developed version of the talk in 9 May, 2008 entitled Numerical invariants from knot groups".
Let $K$ be a knot in $S3$, $G(K)$ the knot group of $K$, and $G'(K)$ the commutator subgroup of $G(K)$. Then an invariant, denoted by $a(K)$, is the minimum number of elements which generate $G'(K)$ normally in $G(K)$. We named the invariant the {\it Ma-Qiu index} or the {\it MQ index} of $K$.

Let $K_{p,q}$ be the connected sum of the $(2, p)$-torus knot and the $(2, q)$-torus knot. Then our main theorem is that the following three statements are equivalent: (1) $\gcd(p, q)=1$ (2) $m(K_{p,q})=1$ (3) $a(K_{p,q})=1$, where $m(K)$ is the Nakanishi index of $K$.
We proved the equivalence of (1) and (2) by a commutative ring theoretical method, and that of (1) and (3) by a combinatorial group theoretical method.

 Speaker： Ryuji Higa (Kobe University) Title： Tangle sum of alternating tangles yielding a splittable link

We consider the problem to decide whether a given diagram represents a splittable link. Menasco introduced crossing ball argument to prove this problem for the case of alternating diagrams. In this talk, we give a survey of crossing ball argument and study what kind of tangle sum of alternating tangles yielding a splittable link.

 Speaker： Hiromasa Moriuchi (OCAMI, Osaka City University) Title： An enumeration of non-prime theta-curves and handcuff graphs

We have enumerated all the prime theta-curves and handcuff graphs with up to seven crossings by using algebraic tangles and prime basic theta-polyhedra.　Here, a theta-polyhedron is a connected graph embedded in a 2-sphere, whose two vertices are 3-valent, and the rest are 4-valent. We can obtain theta-curve and handcuff graph diagrams from theta-polyhedra by substituting algebraic tan gles for their 4-valent vertices. We can composite many spatial graphs by using connected sum'' of them. However, for spatial graphs, connected sum'' is not unique. Therefore we improve theta-polyhedra to enumerate non-prime theta-curves and handcuff graphs. In this talk, we enumerate non-prime theta-curves and handcuff graphs.

 Speaker： Koya Shimokawa (Saitama University) Title： Site-specific recombination of DNA and Dehn surgery

The action of site-specific recombinases can be analyzed using the tangle method, where the action is characterized topologically by solving the corresponding tangle equations. In this talk, we will show how results on Dehn surgery on knots can be applied to this study and discuss recent progresses.

 Speaker： Andrei Pajitnov (Univ. de Nantes) Title： On the Morse-Novikov number and the tunnel number of knots

Let K be a knot in the three-sphere. The Morse-Novikov number MN(K) of K is the minimal number of critical points of a regular circle-valued Morse function defined on the complement of K. We prove that MN(K) is less than or equal to twice the tunnel number of the knot and present consequences of this result.

 Speaker： Ayumu Inoue (Tokyo Institute of Technology) Title： Quandle and hyperbolic volume

In this talk, we will show that the hyperbolic volume of a hyperbolic knot is a quandle cocycle invariant. Further we will show that it completely determines invertibility and positive/negative amphicheirality of hyperbolic knots.

 Speaker： Tomomi Kawamura (Graduate school of Mathematics, Nagoya University) Title： An estimate of the Rasmussen invariant for links

Improving the slice-Bennequin inequality shown by Rudolph, we estimate some knot or link invariants, especially the knot invariant defined by Ozsvath and Szabo and the Rasmussen invariant for links introduced by Beliakova and Wehrli.

 Speaker： Takuji Nakamura (Osaka Electro-Communication University) Title： Local moves and Periodic knots

For each local move $T$, we can define the $T$-gordian distance for two knots. Our motivation is to research a relationship between the periodicity of knots and $T$-gordian distance. In this talk, we pay attention to $C_n$-move as $T$. In fact we show that for any $p$ and any $n(>2)$ there exists a periodic knot of period $p$ whose $C_n$-gordian distance to the trivial knot is one.

 Speaker： Shin Satoh (Kobe University) Title： Triviality of 2-knot with three sheets

The sheet number of a $2$-knot is a quantity analogous to the crossing number of a $1$-knot. It is known that the sheet number is equal to one if and only if the $2$-knot is trivial and that there is no $2$- knot of sheet number two. We prove that there is no $2$-knot of sheet number three. Therefore, the sheet number of any non-trivial $2$-knot is at least four. It is an open problem whether there is a $2$-knot of sheet number four other than the spun trefoil and the $2$-twist-spun trefoil.

 Speaker： Taizo KANENOBU (Osaka City University) Title： H(2)-unknotting number of a knot

This is a joint work with Yasuyuki Miyazawa.
An $H(2)$-move is a local move of a knot which is performed by adding a half-twisted band. It is known an $H(2)$-move is an unknotting operation. We define the $H(2)$-{\it unknotting number} of a knot $K$ to be the minimum number of $H(2)$-moves needed to transform $K$ into a trivial knot. We give several methods to estimate the $H(2)$-unknotting number of a knot.

 Speaker： Kanako Oshiro (Hiroshima university, JSPS Research Fellow) Title： Estimate of minimal triple point numbers of surface-links by symmetric quandle cocycles

It is known that minimal triple point numbers of orientable surface-links are estimated by quandle cocycle invariants. In this talk, we introduce symmetric quandle cocycle invariants. By the invariants, we can estimate minimal triple point numbers of non-orientable surface-links.

 Speaker： Kazuhiro Ichihara (Nara University of Education) Title： Seifert fibered surgeries on Montesinos knots (based on joint works with In Dae Jong and Shigeru Mizushima)

In this talk, Dehn surgeries on hyperbolic Montesinos knots yielding Seifert fibered spaces will be discussed. Actually we will consider the following cases;
1) the resultant manifolds have finite fundamental groups,
2) the knots are alternating.
We will give a complete classification of such surgeries in the case (1), and find certain restrictions on the knots and surgeries in the case (2).

 Speaker： Yoshihiro Fukumoto (Tottori University of Environmental Studies) Title： Bounding genera of several infinite families of Brieskorn homology 3-spheres

The bounding genus is a homology cobordism invariant of homology 3-spheres introduced by Y. Matsumoto in 1982 and defines a kind of distance between homology 3-spheres. Matsumoto gave upper bounds of the bounding genera for certain infinite families of homology 3-spheres by handling techniques of the Dehn-Kirby calculus and formulated his celebrated 11/8-conjecture. In this talk, we determine the bounding genera of several infinite families of Brieskorn homology 3-spheres in the list given by Matsumoto. In fact, we give lower bounds in terms of a signature defect-type invariant (w-invariant) by using of a V-manifold version of Furuta's 10/8-inequality, and combining with Matsumoto's estimates we can determine the bounding genera. Thanks to a referee's comment, we could determine the bounding genera for the infinite families by using an equivalence of the w-invariants and the Neumann-Siebenmann invariants. If we have time, I would also like to talk about a possibility of a generalization of bounding genera under the cobordism category of 3-manifolds by using of a V-manifold version of the Furuta-Kametani 10/8-inequality for the case with positive first Betti numbers.

 Speaker： Ryosuke YAMAMOTO (Osaka City University, OCAMI) Title： An invariant of fibered knots from the third Johnson-Morita homomorphism of mapping class group

We introduce a new calculation of Alexander-Conway polynomial of fibered knots in a closed orientable 3-manifold, and we will discuss to build an invariant of fibered knots by using this calculation with the third Johnson-Morita homomorphism of mapping class group.

 Speaker： Teruhisa Kadokami (Department of Applied Mathematics, Dalian University of Technology) Title： Twisted Alexander invariants of groups

The purpose of this talk is to point out that (1) representations into matrix groups are not needed only to define twisted Alexander invariants, and that (2) representing into matrix groups is an effective method to obtain polynomial or function invariants from twisted Alexander matrices.

We notice that (1) and (2) are not contradictory statements.

Let $G$ be a group, and $N$ a normal subgroup of $G$.
To show (1), we define the twisted Alexander module and matrix, which we call twisted Alexander invariants, related with a pair $(N, G)$ without using representations.

Since to obtain the determinant of a matrix with its entries in a non-commutative group ring is very hard (it cannot be done usually without using K-theory), representing the group into matrix groups is effective to obtain the determinant".
This implies (2).

We suggest to reexamine the usual definition of twisted Alexander invariants from this point of view.

 Speaker： Akio KAWAUCHI (Osaka City University) Title： The warping degree and the unknotting number of a spatial graph

In a research of a protein, a molecule, or a polymer, it is important to understand a spatial graph with vertices of degree one such as a knotted arc geometrically and topologically. In this talk, we introduce a geometric invariant of every connected spatial graph which we call the warping degree. The warping degree is meaningful even for a knotted arc. For every connected spatial graph without vertices of degree one, this invariant is used to define two kinds of topological invariants of the graph. One invariant is the minimal warping degree for the isotopy class of the graph and the other invariant is a natural generalization of the unknotting number of a knot, which coincides with the usual unknotting number for every spatial plane graph. We call this invariant the unknotting number of a spatial graph. This unknotting number is generalized to every connected spatial graph. A generalization of the invariants to a disconnected spatial graph is also easily attained.

 Speaker： Toshifumi Tanaka, (Osaka City University Advanced Mathematical Institute) Title： Signed Gordian distance and Jones polynomials

Recently, Y. Nakanishi, T. Hasegawa and S. Yamada have investigated the Gordian distance $d$ between two knots with low crossing number using Alexander matrices, signatures and a result of H. Murakami concerning the Gordian distance between a knot and a two-bridge knot. Then Nakanishi has given a problem to detect $d(5_{2},6_{1})$, $d(5_{2},6_{1}^{*})$, $d(5_{2},6_{2})$, $d(6_{1},6_{2}^{*})$ and $d(6_{1},6_{3})$.
In this talk, first we define the signed Gordian distance and show that both $d(5_{2},6_{1})$ and $d(6_{1},6_{2}^{*})$ are two by using signatures and Jones polynomials.
We also introduce a lower bound for the Gordian distance in terms of the Q-polynomial. Finally, we show a list of results of computations for the Gordian distance for knots with crossing number less than or equal to eight.

 Speaker： Motoo Tange, (RIMS-Research Institute for Mathematical Sciences Kyoto University) Title： The alternating condition by Ozsvath and Szabo, and the wave theorem by Homma, Ochiai, and Takahashi

Ozsvath-Szabo proved that knots yielding lens spaces satisfy an alternating condition of the Alexander polynomials.
On the other hand, Homma, Ochiai and Takahashi proved the wave theorem for genus 2 Heegaard decomposition of $S3$.
The speaker will talk about a relation between the alternating codition and the wave theorem.

 Speaker： Kouichi Yasui, (Graduate School of Science, Osaka University) Title： Elliptic surfaces without 1-handles

Harer-Kas-Kirby conjectured that the elliptic surface $E(1)_{2,3}$ requires a 1-handle. The speaker previously constructed a 4-manifold which has the same Seiberg-Witten invariant as $E(1)_{2,3}$ and has neither 1- nor 3-handles. Recently Akbulut proved that $E(1)_{2,3}$ admits neither 1- nor 3-handles. In this talk, we prove that the elliptic surface $E(n)_{p,q}$ ((p,q)=(2,3),(2,5),(3,4)) has no 1-handles. Our method is different from Akbulut.

 Speaker： Kengo Kishimoto, (Graduate School of Science, Osaka City University) Title： The dealternating number and the alternation number of a closed 3-braid (joint work with Tetsuya Abe (Graduate School of Science, Osaka City University))

This is a joint work with T. Abe.
We give an upper bound for the dealternating number of a closed 3-braid. As an application, we determine the dealternating numbers and the alternation numbers of some closed positive 3-braid knots. We also show that there exist infinitely many positive knots with any dealternating number (or any alternation number) and any braid index.

 Speaker： Kokoro Tanaka, (Department of Mathematics, Tokyo Gakugei University) Title： A categorification of the one-variable Kamada-Miyazawa polynomial (joint work with Atsushi ISHII (RIMS, Kyoto University))

Khovanov homology is a homology theory for classical links whose graded Euler characteristic is the Jones polynomial. If we want to extend Khovanov homology to virtual links, Khovanov's construction does not immediately work and the main difficulty arising is the existence of M\"{o}bius cobordisms (bifurcations of type $1 \rightarrow 1$).
In this talk, we construct an extension of Khovanov homology to virtual links by taking suitable grading shifts and assigning one of two non-zero maps to each of the M\"{o}bius cobordisms. Our homology theory is a categorification of a one-variable specialization of the Kamada-Miyazawa polynomial.

 Speaker： Daniel Moskovich (Osaka City University Advanced Mathematical Institute) Title： Towards surgery presentations of metabelian coloured knots and their covering links

A knot $K$ and a representation $\rho$ from its knot group onto a finite permutation group $G$ together define a 3-manifold $M_{(K,\rho)}$ via the covering space construction. A surgery presentation for $M_{(K,\rho)}$ would in principle allow us to examine invariants of $M_{(K,\rho)}$ and of $(K,\rho)$ which are constructed via surgery, such as analogues to the Alexander polynomial and Casson invariant. We address the problem of finding surgery presentations for $M_{(K,\rho)}$ when $G$ is a metabelian group which is the semidirect product of a cyclic group with a finite number of copies of the cyclic group $Z_p$ for some prime $p$, through an approach based on a choice of band projection of $K$. This approach is effective when $G=D_{2p}=Z_2\times Z_p$ and when $G=A_4=Z_3\times (Z_2 \times Z_2)$. This is work in progress, joint with Andrew Kricker.

 Speaker： In Dae JONG, (Graduate School of Science, Osaka City University) Title： On a characterization of the Alexander polynomials of alternating knots of genus two

We give a family of linear inequalities on the coefficients of the Alexander polynomials of alternating knots of genus two. The family gives one of the best estimations of them. We also give such families for positive knots of genus two, and for homogeneous knots of genus two. One of the problems we are interested in is a characterization of the Alexander polynomial of an alternating knot. As an approach to this problem, we detect whether each of the Alexander polynomials $\Delta(t)=\sum_{i=0}^{4}a_it^i$ with $|a_0| \leq 10$ is realized as that of an alternating knot.

 Speaker： Tetsuya Abe (Graduate School of Science, Osaka City University) Title： On the Turaev genus

This is a joint work with K.~Kishimoto. In 2006, Dasbach et.al. introduced the Turaev genus of a link, which measures how far it is from alternating links. In this talk, we show the Turaev genus is bounded above by the dealternating number. As an application, we show non-alternating knots of eleven or fewer crossings except two knots have Turaev genus one.

 Speaker： Tamas Kalman (Graduate School of Mathematical Sciences, The University of Tokyo) Title： The problem of maximum Thurston-Bennequin number for knots

Legendrian submanifolds of contact 3-manifolds are one-dimensional, just like knots. This coincidence'' gives rise to an interesting and expanding intersection of contact and symplectic geometry on the one hand and classical knot theory on the other. As an illustration, we will survey recent results on maximizing the Thurston--Bennequin number (which is a measure of the twisting of the contact structure along a Legendrian) within a smooth knot type. In particular, we will indicate how Kauffman's state circles can be used to solve the maximization problem for so-called +adequate (among them, alternating and positive) knots and links.

 Speaker： Teruhisa Kadokami (Department of Applied Mathematics, Dalian University of Technology) Title： Numerical invariants from knot groups (joint work with Ruifeng Qiu (Dalian University of Technology) and Zhiqing Yang (Dalian University of Technology))

In 2006, Jiming Ma and Ruifeng Qiu defined a numerical invariant for a knot in S^3.(Chin.Ann. Math. 27)
Let K be a knot in S^3, G the group of K, and G' the commutator subgroup of G.
An invariant, denoted by a(K), is the minimal number of elements which generate G' normally in G.
They showed the following results:

Theorem [Ma-Qiu]
(1) m(K) \le a(K) \le u(K),
(2) max{a(K_1), a(K_2)} \le a(K_1#K_2) \le a(K_1)+a(K_2), where m(K) is the Nakanishi index of K, u(K) is the unknotting number of K, and K_1# K_2 is the connected sum of knots K_1 and K_2.

We showed the following:

Main Theorem a(K) \le r(K)-1, where r(K) is the rank of K.

By combining with known results, we have:

Corollary m(K) \le a(K) \le min{u(K), t(K)}, where t(K) is the tunnel number of K.

We remark that a(K) \ge 1 for a non-trivial knot K.
We would like to use the invariant a(K) for additivity problems of u(K) and t(K) as a lower bound.

 Speaker： Ikuo Tayama (Osaka City University Advanced Mathematical Institute) Title： A complete classification of 3-manifolds with lengths up to 9 by a canonical order

This is a joint work with A. Kawauchi. A well-order was introduced on the set of links by A. Kawauchi. This well-order also naturally induces a well-order on the set of prime link exteriors and eventually induces a well-order on the set of closed connected orientable 3-manifolds. With respect to this order, we enumerated the prime links with lengths up to 10 and the prime link exteriors with lengths up to 9. In this talk, we show a complete list of 3-manifolds with lengths up to 9 by using the enumeration of the prime link exteriors.

 Speaker： Kazuhiro Sakuma (Kinki University) Title： A question on the special values of Alexander polynomials which detect the smooth structure on spheres

J. Levine proved that the Kervaire invariant of a (4m+1)- dimensional homotopy sphere was completely determined by the special value of the Alexander polynomial. This can be reformulated by using the Legendre (Jacobi) symbol based on the quadratic residue laws due to Gauss. Here is a question; What is the algebraic essence of this fact from number theoretical viewpoint?''

 Speaker： Masahide Iwakiri (Osaka City University Advanced Mathematical Institute) Title： Calculation of quandle cocycle invariants of Suzuki's $\theta_n$-curves

In joint work with A. Ishii, we define quandle colorings and quandle cocycle invariants for handlebody-links and spatial graphs. In this talk, we introduce the invariants shortly, and calculate them for Suzuki's $\theta$_n-curves.

 Speaker： Toshifumi Tanaka (Osaka City University Advanced Mathematical Institute) Title： The quasipositivation number of a knot

All quasipositive links can be realized as transverse intersections of complex plane curves with the standard sphere, and the converse is also known to be true if defining polynomials are non-constant. In this talk we study the Gordian distance from a knot to the set of quasipositive knots that we call the quasipositivation number. In general, it is not easy to detect even quasipositivity. We estimate the number by using the Rasmussen invariant.