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Speaker ：Selman Akbulut (Michigan State University, USA) Title ：Corks, Palfs, exotic structures on 4-manifolds （Abstract） （PDF） Date ：February 2 (Fri.) 15：00～16：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Akira Yasuhara (Tokyo Gakugei University) Title ：On classifications of links up to $C_n$-moves （Abstract） （PDF） Date ：January 26 (Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Yasuyuki Miyazawa (Department of Mathematical Sciences, Yamaguchi University) Title ：A variety of virtual link polynomial with multiple variables （Abstract） （PDF） Date ：January 19 (Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Ikuo Tayama (Osaka City University) Title ：Enumerating 3-manifolds by a canonical order II （Abstract） （PDF） Date ：January 19 (Fri.) 15：00～16：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Hiromasa Moriuchi (Graduate School of Science, Osaka City University) Title ：Enumerations of theta-curves and handcuff graphs （Abstract） （PDF） Date ：January 12 (Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Fumikazu Nagasato (Department of Mathematics Tokyo Institute of Technology) Title ：Algebraic equations and knot invariants （Abstract） （PDF） Date ：January 12 (Fri.) 15：00～16：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Hirotaka AKIYOSHI (Osaka City University Advanced Mathematical Institute) Title ：Volume of the convex core of a punctured torus group （Abstract） （PDF） Date ：December 1 (Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Atsushi ISHII (Graduate School of Science, Osaka University) Title ：The pole diagram and the Miyazawa polynomial （Abstract） （PDF） Date ：November 10 (Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Takuji NAKAMURA (Research Center for Physics and Mathematics, Faculty of Engineering, Osaka Electro-Communication University) Title ：Notes on Futer-Purcell's inequality for genera of knots and hyperbolic knots with trivial Alexander polynomial （Abstract） （PDF） Date ：November 24 (Fri.) 15：00～16：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Seiichi KAMADA (Department of Mathematics, Hiroshima University) Title ：Quandles with good involutions and their homologies （Abstract） （PDF） Date ：November 10 (Fri.) 17：00～18：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Hiroko MURAI (Graduate School of Humanities, Nara Women's University) Title ：Gap of the depths of adjacent leaves of finite depth foliations （Abstract） （PDF） Date ：November 10 (Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Taizo Kanenobu (Department of Mathematics, Osaka City University) Title ：The block numbers of 2-bridge knots and links （Abstract） （PDF） Date ：October 27(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Makoto Ozawa （Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University Title ：A property of diagrams of the trivial knot （Abstract） （PDF） Date ：October 20(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Alexander Stoimenow (Research Institute for Mathematical Sciences Kyoto University, COE Program Research Fellow) Title ：Bennequin surfaces and braid index of alternating knots （Abstract） （PDF） Date ：October 13(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Taizo Kanenobu (Department of Mathematics, Osaka City University) Title ：Finite-Type Invariants of Order 4 for Oriented 2-Component links （Abstract） （PDF） Date ：July 7(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Teruhisa Kadokami (Osaka City University (OCAMI) COE Researcher) Title ：On the Alexander polynomial satisfying Ozsv\'ath-Szab\'o's condition for lens sugery （Abstract） （PDF） Date ：June 30(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Nafaa Chbili （COE research member, OCAMI） Title ：Toward an equivariant Khovanov homology （Abstract） （PDF） Date ：June 30(Fri.) 14：00～15：30 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Toshifumi Tanaka (Osaka City University (OCAMI) COE Fellow) Title ：On slice knots in 4-manifolds （Abstract） （PDF） Date ：June 23(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Tatsuya Tsukamoto　(Waseda University JSPS Research Fellow) Title ：Special positions for spanning surfaces in link complements （Abstract） （PDF） Date ：June 16(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Reiko Shinjo (OCAMI, COE Researcher) Title ：An infinite sequence of non conjugate $4$-braids representing the same knot of braid index 4 （Abstract） （PDF） Date ：May 26(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Gwenael Massuyeau (CNRS - Louis Pasteur University, Strasbourg, CNRS researcher) Title ：Some finiteness properties for the Reidemeister-Turaev torsion of three-manifolds. （Abstract） （PDF） Date ：May 19(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Yo'av Rieck (University of Arkansas, Assistant Professor) Title ：The growth rate of tunnel numbers of m-small knots （Abstract） （PDF） Date ：May 12(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Masahide Iwakiri 　(Department of Mathematics, Hiroshima University, JSPS Research Fellow) Title ：Unknotting singular surface braids by crossing changes （Abstract） （PDF） Date ：April 28(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Ryosuke Yamamoto (OCAMI COE Researcher) Title ：Overtwisted open books and Stallings twist （Abstract） （PDF） Date ：April 21(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top Speaker ：Kenichi Fujiwara (Department of Mathematics, Tokyo Institute of Technology) Title ：Refined Kirby calculus for rational homology spheres of prime orders （Abstract） （PDF） Date ：April 14(Fri.) 16：00～17：00 Place ：Dept. of Mathematics, Sci. Bldg., 3153 Top

Speaker：	Selman Akbulut (Michigan State University, USA)
Title：	Corks, Palfs, exotic structures on 4-manifolds

One way to understand 4-manifolds is to decompose them it into small understandable pieces (Corks), and make these pieces symplectic Lefschetz fibrations (Palfs). We will survey these results, and as an example discuss such decomposition of an exotic rational surface.

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Speaker：	Akira Yasuhara (Tokyo Gakugei University)
Title：	On classifications of links up to $C_n$-moves

A {\em $C_n$-move} ($n\in{\Bbb N}$) is a local move on links defined by Habiro, which can be regarded as a `higher order crossing change'. The {\em $C_n$-equivalence} is an equivalence relation on links generated by $C_n$-move. The $C_m$-equivalence implies the $C_n$-equivalence for $m>n$. So the {\em $C_n$-classification}, which is the classification up to $C_n$-equivalence, of links becomes finer as $n$ increases. The $C_2$-classification of links and the $C_3$-classification of links with 2 or 3 components, or of algebraically split links are known. Here we give several classifications of certain sets of links by using Milnor invariants.

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Speaker：	Yasuyuki Miyazawa (Department of Mathematical Sciences, Yamaguchi University)
Title：	A variety of virtual link polynomial with multiple variables

The speaker introduced a virtual link polynomial with multiple variables in the workshop "Tohoku Knot Seminar" at Zao in Yamagata in late autumn of 2006. In this talk, we construct a variety of multi-variable polynomial for virtual links by adding the concept of a weight map to the definition of the above polynomial. We also refer to a relationship between the new polynomial and a polynomial invariant defined by the speaker before.

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Speaker：	Ikuo Tayama (Osaka City University)
Title：	Enumerating 3-manifolds by a canonical order II

This work is a joint work with A. Kawauchi.
A well-order (called a {\it canonical order}) was introduced on the set of links by A. Kawauchi [K].
This well-order 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.

We assign to every link a lattice point whose length is equal to the minimal crossing number on closed braid forms of the link. We call this number the {\it length} of the link. We note that a link $L$ is smaller than a link $L'$ in the canonical order if the length of $L$ is smaller than that of $L'$. We define the {\it length} of a prime link exterior as the minimal length of a prime link whose exterior is homeomorphic to the given prime link exterior and we define the {\it length} of a closed connected orientable $3-$manifold is the minimal length on prime link exteriors realizing the $3-$manifold as the $0$ surgery manifold along the prime link.

With respect to the canonical order, we enumerated the prime links with up to length $10$ [KT1] and the prime link exteriors with up to length $9$ [KT2]. We are now enumerating the $3-$manifolds with up to length $9$. We classify the manifolds according to their first homology groups. There are 10 types of groups $0,{\bf Z},{\bf Z}\oplus {\bf Z},{\bf Z}\oplus {\bf Z}\oplus {\bf Z}, {\bf Z}\oplus {\bf Z}_2\oplus {\bf Z}_2, {\bf Z}_2,{\bf Z}_2\oplus {\bf Z}_2, {\bf Z}_3\oplus {\bf Z}_3,{\bf Z}_4,{\bf Z}_4\oplus {\bf Z}_4$ and we have respectively 16,62,16,4,5,7,15,7,5,5 links with these types of groups. We enumerated the manifolds with the group equal to ${\bf Z}$ in [KT3]

In this talk, we enumerate the manifolds with the group equal to ${\bf Z}\oplus {\bf Z},{\bf Z}\oplus {\bf Z}\oplus {\bf Z}, {\bf Z}_2,{\bf Z}_2\oplus {\bf Z}_2, {\bf Z}_4,{\bf Z}_4\oplus {\bf Z}_4$..

{\bf References}

[K] A. Kawauchi, A tabulation of 3-manifolds via Dehn surgery, Boletin de la Sociedad Matematica Mexicana (3) 10 (2004), 279--304.
[KT1] A. Kawauchi and I. Tayama, Enumerating prime links by a canonical order, Journal of Knot Theory and Its Ramifications Vol. 15, No. 2 (2006) 217--237
[KT2] A. Kawauchi and I. Tayama, {\it Enumerating the exteriors of prime links by a canonical order}, in: Proc. Second East Asian School of Knots, Links, and Related Topics (Darlian, Aug. 2005), to appear.
[KT3] A. Kawauchi and I. Tayama, {\it Enumerating $3-$manifolds by a canonical order}, in: Proc. of ILDT (Hiroshima Univ., July 2006), to appear.

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Speaker：	Hiromasa Moriuchi (Graduate School of Science, Osaka City University)
Title：	Enumerations of theta-curves and handcuff graphs

We enumerate all the $\theta$-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 2-sphere, whose two vertices are 3-valent, and the others are 4-valent. There exist twenty-four prime basic $\theta$-polyhedra with up to seven 4-valent vertices. We can obtain a $\theta$-curve or handcuff graph diagram from a prime basic $\theta$-polyhedron by substituting algebraic tangles for their 4-valent vertices.

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Speaker：	Fumikazu Nagasato (Department of Mathematics Tokyo Institute of Technology)
Title：	Algebraic equations and knot invariants

In this talk, for a knot $K$ in 3-sphere $S3$, we define algebraic varieties $\mathcal{F}^{(d)}(K)$ ($d=1,2,3$) in a complex space $\mathbb{C}^{N}$ in the following steps. For a braid presentation $\sigma$ of a knot $K$, we first construct finitely many polynomials $\{p_{\sigma, i}\}_i$ on $\mathbb{C}^{N}$ by using an action of the braid $\sigma$ on the Kauffman bracket skein module (KBSM) of a handlebody at $t=-1$ with {\it trace-free condition}. Then the ideal $\mathcal{SL}^{(3)}(\sigma)$ generated by the polynomials $\{p_{\sigma,i}\}_i$ gives an algebraic variety $\mathcal{F}^{(3)}(\sigma)$ via the Hilbert Nullstellensatz. In fact, $\mathcal{F}^{(3)}(\sigma)$ turns out to be invariant under the Markov moves and thus becomes a knot invariant. This is a desired variety $\mathcal{F}^{(3)}(K)$. The above process can be used for {\it restrictions} $\mathcal{SL}^{(2)}(\sigma)$ and $\mathcal{SL}^{(1)}(\sigma)$ of the ideal $\mathcal{SL}^{(3)}(\sigma)$. Then we can get knot invariants $\mathcal{F}^{(d)}(K)$ ($d=1,2$).

The first variety $\mathcal{F}^{(1)}(K)$ is actually trivial invariant. The third one $\mathcal{F}^{(3)}(K)$ can be considered　as a variety containing ``a section'' of the $SL(2,\mathbb{C})$- character variety of the knot group by using Bullock's theorem (quantization of the $SL(2,\mathbb{C})$-character variety). This view point gives relationships of the variety $\mathcal{F}^{(3)} (K)$　with the number of $SL(2,\mathbb{C})$-irreducible metabelian characters　of the knot group (the knot determinant), and moreover the maximal degree (or {\it span}) of the A-polynomial $A_K(m,l)$ in terms of $l$, which polynomial is a knot invariant introduced by Cooper, Culler, Gillet, Long and Shalen. Regarding the second variety $\mathcal{F}^{(2)}(K)$, the quotient ring $\mathbb{C}[x_1,\cdots,x_n]/\mathcal{SL}^{(2)} (\sigma)$　($n\leq N$) turns out to be isomorphic to the degree $0$ knot contact homology which was researched by L. Ng in detail.

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Speaker：	Hirotaka AKIYOSHI (Osaka City University Advanced Mathematical Institute)
Title：	Volume of the convex core of a punctured torus group

For any quasifuchsian punctured torus group, the Weil-Petersson distance between the conformal structure of the boundary Riemann surface and the Jorgensen's side parameter is bounded above by a universal constant. J. Brock showed that the volume of the convex core of a quasifuchsian manifold is quasi-comparable to the Weil-Petersson distance between the conformal structures of the boundary Riemann surfaces. Combining these theorems, the volume of the convex core of a quasifuchsian punctured torus group is estimated by a combinatorial structure of the Ford domain of the group.

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Speaker：	Atsushi ISHII (Graduate School of Science, Osaka University)
Title：	The pole diagram and the Miyazawa polynomial

We introduce the pole diagram, which helps to retrieve information from a knot diagram when we smooth crossings. By using the notion, we define a bracket polynomial for the Miyazawa polynomial. The bracket polynomial gives a simple definition and evaluation for the Miyazawa polynomial. Then we show that the virtual crossing number of a virtualized alternating link is determined by its diagram.

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Speaker：	Takuji NAKAMURA (Research Center for Physics and Mathematics, Faculty of Engineering, Osaka Electro-Communication University)
Title：	Notes on Futer-Purcell's inequality for genera of knots and hyperbolic knots with trivial Alexander polynomial

The genus of a knot is an intuitive geometric invariant for knots. However it is hard to determine in general. It is well-known that the degree of the Alexander polynomial of a knot estimates the genus of the knot from below. Recently, Futer-Purcell showed that if a diagram of a link satisfies certain several conditions then the link is hyperbolic and the genus of the link is estimated from below by some complexity of a diagram. In this talk, we introduce Futer-Purcell's inequality and construct a hyperbolic knot of higher genus but whose Alexander polynomial is trivial by using the inequality.

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Speaker：	Seiichi KAMADA (Department of Mathematics, Hiroshima University)
Title：	Quandles with good involutions and their homologies

Quandles and their homologies are used to construct invariants of oriented links or oriented surface-links in 4-space. On the other hand the knot quandle can still be defined in the case where the links or surface-links are not oriented, but in this case it cannot be used to construct homological invariants. Here we introduce the notion of a quandle with a good involution, and its homology groups. We can use them to construct invariants of unoriented links and unoriented, or non-orientable, surface-links in 4-space.

A sketch was given at the conference ``Intelligence of Low Dimensional Topology 2006''.

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Speaker：	Hiroko MURAI (Graduate School of Humanities, Nara Women's University)
Title：	Gap of the depths of adjacent leaves of finite depth foliations

Depth is one of the well-known invariants of codimension one foliations. Roughly speaking, depth is a quantity which describes how far from a fiber bundle structure the foliation is. In this talk, we introduce a quantity called \lq\lq gap\rq\rq of the foliation to deal with behaviors of depths of leaves. More precisely, for a depth $k(\geq 1)$ leaf of a foliation $\mathcal{F}$, we know by the definition of depth of leaves that there exists a depth $k-1$ leaf in $\overline{L}\setminus L $. However, for a leaf $L$ of $\mathcal{F}$ which is not at the maximal depth in $\mathcal{F}$, it is not necessary the case that there exists a leaf $L^\prime$ at depth $(\textrm{depth}(L)+1)$ such that $L\subset\overline{L^\prime}\setminus L^ \prime$. In this case, there is a \lq\lq gap\rq\rq\ between the depth of $L$ and depths of the adjacent leaves. Roughly speaking, the gap of $\mathcal{F}$ is the maximal value of the gaps between the depths of the leaves of $\mathcal{F}$. As an application, by using this invariant, we give an estimation of depth of foliations of the manifolds which we considered in [M].

[M] H. Murai,
{\it Depths of the Foliations on 3-Manifolds Each of Which Admits Exactly One Depth 0 Leaf}, J. Knot Theory Ramifications, to appear.

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Speaker：	Taizo Kanenobu (Department of Mathematics, Osaka City University)
Title：	The block numbers of 2-bridge knots and links

For a $2$-bridge knot or link, we introduce a new invariant, the {\it block number} from the Conway presentation. We can determine the block number by either the Kauffman polynomial, the Brandt-Lickorish-Millett-Ho's $Q$ polynomial, or the Jones polynomial.

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Speaker：	Makoto Ozawa （Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University
Title：	A property of diagrams of the trivial knot

We give a necessary condition for a diagram to represent the trivial knot.
A preprint is available from http://arxiv.org/abs/math.GT/0606293.

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Speaker：	Alexander Stoimenow (Research Institute for Mathematical Sciences Kyoto University, COE Program Research Fellow)
Title：	Bennequin surfaces and braid index of alternating knots

The braid index inequality of Morton-Williams-Franks is exact for many alternating knots, but Murasugi gave an example of inexact inequality of crossing number 18, genus 6 and braid index 6. In my talk I will explain the proof of exactness of the Morton-Williams-Franks inequality for alternating knots of

1. at most 18 crossings (except Murasugi's example and its mutant)
2. genus at most 4 and
3. braid index at most 4 (actually Morton-Williams-Franks bound at most 4, and here links could be included)

I will explain how to extend these proofs to show that many alternating knots have a minimal string Bennequin surface (= minimal genus braided Seifert surface), in particular alternating knots of genus at most 3 or at most 16 crossings.

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Speaker：	Taizo Kanenobu (Department of Mathematics, Osaka City University)
Title：	Finite-Type Invariants of Order 4 for Oriented 2-Component links

We express a basis for the space of finite type invariants of order less than or equal to four for two-component links in terms of the Conway polynomial, the linking number, and the HOMFLYPT polynomial.
As an application, we give some formulas relating to the HOMFLYPT and Kauffman polynomials.

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Speaker：	Teruhisa Kadokami (Osaka City University (OCAMI) COE Researcher)
Title：	On the Alexander polynomial satisfying Ozsv\'ath-Szab\'o's condition for lens sugery

P. Ozsv\'ath and Z. Szab\'o [3] showed the following:

[Theorem]
Let $K$ be a knot in $S3$.
If $K$ yields a lens space, then the Alexander polynomial of $K$ has the following form:
$${\mit \Delta}_K(t)=(-1)^m+\sum_{i=1}^m(-1)^{m-i}(t^{c_i}+t^{-c_i}) \quad (c_0=0<c_1<c_2<\cdots <c_m)$$

Let $T(r, s)$ be an $(r, s)$-torus knot.
The Alexander polynomial of $T(r, s)$ satisfies Ozsv\'ath-Szab\'o's condition above.
For a positive integer $n$, ${\mit \Delta}_{T(r, s)}(t^n)$ also satisfies the condition.
We showed the following:

[Main Theorem]
Let $K$ be a knot in $S3$.
If $K$ yields a lens space and ${\mit \Delta}_K(t)={\mit \Delta}_{T (r, s)}(t^n)$ where $n$ is a positive integer, then we have $n=1$.

This theorem implies that Ozsv\'ath-Szab\'o's condition does not characterize the Alexander polynomial of a knot in $S3$ having a lens surgery.

[References]
[1] T. Kadokami,
On the Alexander polynomial satisfying Ozsv\'ath-Szab\'o's condition for lens sugery, preprint (2006).

[2] T. Kadokami and Y. Yamada,
A deformation of the Alexander polynomials of knots yielding lens spaces, preprint (2006).

[3] P. Ozsv\'ath and Z. Szab\'o,
On knot Floer homology and lens space surgeries, Topology,44 (2005), 1281--1300.

[4] M. Tange,
Ozsv\'ath Szab\'o's correction term and lens surgery, preprint (2006).

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Speaker：	Nafaa Chbili （COE research member, OCAMI）
Title：	Toward an equivariant Khovanov homology

In this talk, we construct an equivariant Khovanov homology, with coefficients in the finite field $\F_2$, associated to link diagrams with $\Z_p$-symmetry. Then we prove that this homology is conserved under equivariant Reidemeister moves. This equivariant Khovanov homology is better described using the categorification of the Kauffman bracket skein module of the solid torus.

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Speaker：	Toshifumi Tanaka (Osaka City University (OCAMI) COE Fellow)
Title：	On slice knots in 4-manifolds

Let $M$ be a closed oriented 4-manifold and $L$ be an $n$-component link in $M-{\rm int}B4$. $L$ is called a {\it topologically slice link} in $M$ if $L$ bounds $n$ topologically embedded flat 2-disks in $M-{\rm int}B4$. A $1$-component topologically slice link is called {\it topologically slice knot}. For example, every knot is a topologically slice knot in $S2\times S2$ and every knot with trivial Alexander polynomial is a topologically slice knot in $S4$. However, every knot with nontrivial signature is not a topologically slice knot in $S4$.
In this talk, we show that a punctured $M$ admits at least two smooth strucutures if there exists a topologically slice knot which is not a slice knot in $M$. As a corollary, we show that the punctured $CP2$ admits at least two smooth structures.

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Speaker：	Tatsuya Tsukamoto　(Waseda University JSPS Research Fellow)
Title：	Special positions for spanning surfaces in link complements

We study links using the crossing-ball technique of W.Menasco and define special positions for spanning surfaces in link complements. We show that if a given spanning surface is in special position, then the boundary of a neighborhood of the surface is in standard position. Thus we can work on a closed surface in the link complement instead of working on a surface with boundary. We also mension that if a link admits an almost alternating diagram, then we can cut its spanning surface to be in special position.

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Speaker：	Reiko Shinjo (OCAMI, COE Researcher)
Title：	An infinite sequence of non conjugate $4$-braids representing the same knot of braid index 4

Before, we showed the following: For any knot represented as a closed n-braid ($n \ge 3$), there exist an infinite sequence of pairwise non conjugate ($n+1$)-braids representing the knot. Using the similar technique, for some knots of braid index $4$ we construct such an infinite sequence of pairwise non conjugate $4$- braids. As a consequence, we verify that M. Hirasawa's candidate of such a sequence of braids is actually an infinite sequence.

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Speaker：	Gwenael Massuyeau (CNRS - Louis Pasteur University, Strasbourg, CNRS researcher)
Title：	Some finiteness properties for the Reidemeister-Turaev torsion of three-manifolds.

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

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Speaker：	Yo'av Rieck (University of Arkansas, Assistant Professor)
Title：	The growth rate of tunnel numbers of m-small knots

Let $K \subset M$ be a knot in a closed orientable 3-manifold. Let $nK$ be the connected sum of $n$ copies of $k$, $E(\cdot)$ knot exteriors, and $g(\cdot)$ Heegaard genus. In ``On the growth rate of tunnel number of knots'' (to appear in {\it Journal fur die reine und angewandte Mathematik}, available at {\verb http://arxiv.org/abs/math.GT/ 0402025 }) we study the asymptotic behavior of the tunnel number under repeated connected sum operation. We define the growth rate of the tunnel number of $K$ is defined to be:

$$gr_t(k) = \limsup_{n \to \infty} \frac{t(nK) - n t(K)}{n-1}.$$

Let $g = g(E(K)) - g(M)$. Given an integer $i$ ($1 \leq i \leq g$) let $b_i$ be the bridge index of $K$ with respect to Heegaard surfaces of $M$ of genus $g(E(K))-i$. In this talk we prove that if $K$ is meridionally small (that is, the exterior of $K$ admits no essential meridional surface) than the growth rate of the tunnel number of $K$ is:

$$gr_t(K) = \min_{1 \leq i \leq g} 1 - \frac{i} {b_i}.$$

The tools necessary (the strong Hopf--Haken annulus theorem and the Swallow Follow Torus Theorem) will be discussed, as well as some corollaries. Some of the material presented here is still in preparation; some appears in ``Heegaard genus of the connected sum of m-small knots'' (to appear in {\it Communication in Analysis and Geometry}, available at \verb http://arxiv.org/abs/math.GT/0503229 ).

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Speaker：	Masahide Iwakiri (Department of Mathematics, Hiroshima University, JSPS Research Fellow)
Title：	Unknotting singular surface braids by crossing changes

C. A. Giller proved that crossing change is an unknotting operation for surfaces in 4-space. In this talk, we present such an unknotting theorem for singular surface braids, which is given when they have no branch points by S. Kamada. As a consequence, we have Giller's unknotting theorem. Recently, K. Tanaka gave a different proof of our main result.

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Speaker：	Ryosuke Yamamoto (OCAMI COE Researcher)
Title：	Overtwisted open books and Stallings twist

We discuss a characterization of "overtwisted open books" on a closed oriented $3$-manifold, i.e., open book decompositions corresponding to overtwisted contact structures via the Giroux's one-to-one correspondence. We focus on a simple closed curve on fiber surface of an open book along which one can perform Stallings twist, and see that a given open book is overtwisted if and only if it is equivalent to an open book with Stallings twist up to positive stabilization, i.e., plumbing a positive Hopf band to the fiber surface.

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Speaker：	Kenichi Fujiwara (Department of Mathematics, Tokyo Institute of Technology)
Title：	Refined Kirby calculus for rational homology spheres of prime orders

Every integral homology 3-sphere is presented by a framed link whose components have surgery coefficients +-1 and linking numbers 0. By using such framed links and the special Kirby move due to K. Habiro, we can refine Kirby calculus. In this talk, we aim to extend Habiro's refinement to rational homology 3-spheres of prime orders.

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