The 21st Century COE Program Constitution of wide-angle mathematical basis focused on knots Department of Mathematics and Physics Graduate School of Science Osaka City University
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 Friday Seminar on Knot Theory（2007） (2006) (2008)
Organizer in 2007 : TANAKA, Toshifumi
Speaker ：Masahiro Kawami (OCAMI, COE Reseacher)
Title ：Mod4 quadratic forms and diffeomorphisms on non-orientable surfaces
（Abstract） （PDF）
Date ：March 28 (Fri.) 15：00～16：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Hirotaka Akiyoshi (Osaka City University Advanced Mathematical Institute)
Title ：Side parameter for the punctured torus groups
（Abstract） （PDF）
Date ：February 29 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Takuji Nakamura (Osaka Electro-Communication University)
Title ：On knots of Delta unknotting number one from a view of the positivity for knots
（Abstract） （PDF）
Date ：February 22 (Fri.) 16：30～17：30
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Alexander Stoimenow (OCAMI, COE fellow)
Title ：Vassiliev invariants, Seifert matrix, and hyperbolic volume of knots
（Abstract） （PDF）
Date ：February 22 (Fri.) 15：30～16：30
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Hiromasa Moriuchi (OCAMI, COE researcher)
Title ：Classifications of theta-curves and handcuff graphs
（Abstract） （PDF）
Date ：February 8 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Thomas Mattman (California State University)
Title ：Boundary Slope Diameter and Crossing Number of 2-Bridge Knots
（Abstract） （PDF）
Date ：February 1 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Taizo Kanenobu (Osaka City University)
Title ：The sharp-unknotting number of a torus knot
（Abstract） （PDF）
Date ：December 14 (Fri.) 13：30～14：30
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Ikuo Tayama (OCAMI, COE researcher)
Title ：Enumerating 3-manifolds with lengths up to 9 by a canonical order
(joint work with Akio Kawauchi (Osaka City University))
（Abstract） （PDF）
Date ：November 30 (Fri.) 15：30～16：30
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Daniel Moskovich
Title ：Two Surgery Presentations for Dihedral Covering Spaces
(joint work with Andrew Kricker (Nanyang Technological University))
（Abstract） （PDF）
Date ：November 30 (Fri.) 14：00～15：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Noboru Ito (Waseda University)
Title ：Invariants via word for curves
（Abstract） （PDF）
Date ：November 16 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Yo'av Rieck (University of Arkansas)
Title ：On the Heegaard genus of knot exteriors
(joint with Tsuyoshi Kobayashi (Nara Women's University))
（Abstract） （PDF）
Date ：November 9 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Teruhisa Kadokami (OCAMI, COE researcher)
Title ：Calculating the Casson-Walker invariants of cyclic branched coverings over knots yielding lens spaces
(a joint work with Yasuyoshi Tsutsumi (Oshima National College of Maritime Technology)
and Yukihiro Tsutsumi (Sophia University))
（Abstract） （PDF）
Date ：October 12(Fri.) 15：00～16：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Yoshihiro Fukumoto (Tottori University of Environmental Studies)
Title ：Homology spin cobordism problem of plumbed 3-manifolds and the cup product structures
（Abstract） （PDF）
Date ：October 12 (Fri.) 14：00～15：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Takao Satoh
(Department of Mathematics Graduate School of Science, Osaka University)
Title ：Twisted homology groups of the automorphism group of a free group knots
（Abstract） （PDF）
Date ：July 13 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Toshifumi Tanaka (OCAMI, COE fellow)
Title ：Maximal Thurston-Bennequin numbers and Rasmussen invariants of doubled knots
（Abstract） （PDF）
Date ：July 6 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Ushijima Akira
(Graduate School of Natural Science and Technology, Kanazawa University)
Title ：Hyperbolic spatial graphs coming from strongly invertible knots
(joint work with Kazuhiro Ichihara (Nara University of Education))
（Abstract） （PDF）
Date ：June 29 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Atsushi Ishii (Research Institute for Mathematical Sciences, Kyoto University)
Title ：The skein index for link invariants
（Abstract） （PDF）
Date ：June 15 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Shin Satoh (Department of Mathematics, Kobe University)
Title ：The sheet numbers of 2-knots
（Abstract） （PDF）
Date ：June 8 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Iwakiri Masahide (OCAMI, COE fellow)
Title ：Quandle cocycle invariants of charts with six white vertices
（Abstract） （PDF）
Date ：June 1 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Motoo Tange
(Department of Mathematics Graduate School of Science, Osaka University)
Title ：On tight contact structure and lens surgery
（Abstract） （PDF）
Date ：May 11 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Makiko Ishiwata (Hirasawa) (OCAMI, COE researcher)
Title ：A classification of links up to 5-move equivalence
（Abstract） （PDF）
Date ：April 27 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：YAMAMOTO, Ryosuke (OCAMI, COE researcher)
Title ：Complexity of open book decompositions via arc complex
(Joint work with Toshio Saito (Nara Women's University))
（Abstract） （PDF）
Date ：April 20 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Kouichi Yasui (Osaka University)
Title ：Small exotic rational surfaces without 1- and 3-handles
（Abstract） （PDF）
Date ：April 13 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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 Abstracts

 Speaker： Masahiro Kawami (OCAMI, COE Reseacher) Title： Mod4 quadratic forms and diffeomorphisms on non-orientable surfaces

We investigate Mod4 quadratic forms on the $\mathbb{Z}_{2}$-coefficient first homology group of a non-orientable surface. In the case of the genus of a surface is small, we can enumerate them up to isomorphims easily and they are few. In this talk, we study the self-diffeomorphims on a non-orientable surface which leave a given Mod4-quadratic form invariant.

 Speaker： Hirotaka Akiyoshi (Osaka City University Advanced Mathematical Institute) Title： Side parameter for the punctured torus groups

The side parameter for the quasifuchsian punctured torus groups is defined by Jorgensen in his famous unfinished paper on the combinatorial structures of the Ford domains of such groups. The side parameter is a complete invariant on the quasifuchsian space, and is extended to an invariant on the closure of the space. The main results of this talk are the following:
1. The extended side parameter is also a complete invariant.
2. The composition of the inverse of the end invariant map and the side parameter map is a self-homeomorphism on the common target space of the invariants.

 Speaker： Takuji Nakamura (Osaka Electro-Communication University) Title： On knots of Delta unknotting number one from a view of the positivity for knots

A Delta unknotting number for a knot is the minimal number of Delta moves" needed to create the unknot. A knot is said to be positive if it has a diagram whose all crossings are positive. In this talk, we show that a Delta unknotting number one positive knot is the right-handed trefoil knot. We also observe almost positive knots of Delta unknotting number one.

 Speaker： Alexander Stoimenow (OCAMI, COE fellow) Title： Vassiliev invariants, Seifert matrix, and hyperbolic volume of knots

Given any knot K, we construct hyperbolic knots with arbitrarily large volume, with the same Seifert matrix and the same Vassiliev invariants of a bounded degree as K. An additional feature of our knots is that they have bounded free genus.

 Speaker： Hiromasa Moriuchi (OCAMI, COE researcher) Title： Classifications of theta-curves and handcuff graphs

We have enumerated all the theta-curves and handcuff graphs with up to seven crossings. To make a table of them, we need some spatial graph invariants.
In 1989, S. Yamada defined the topological invariant of spatial graphs known as the Yamada polynomial. In this talk, we compute the Yamada polynomial of some spatial graphs, and mention some properties of the Yamada polynomial.

 Speaker： Thomas Mattman (California State University) Title： Boundary Slope Diameter and Crossing Number of 2-Bridge Knots

In joint work with Maybrun and Robinson, we prove that for 2-bridge knots, the diameter of the set of boundary slopes is twice the crossing number. After some topological preliminaries, the argument quickly becomes combinatorial and is based on comparing various continued fraction representations of the same fraction.

 Speaker： Taizo Kanenobu (Osaka City University) Title： The sharp-unknotting number of a torus knot

The sharp-unknotting number was introduced by Hitoshi Murakami. He estimates for the sharp-unknotting number from below using the signature, Arf invariant, homology invariant from a cyclic covering. We give an estimation for the sharp-unknotting numbers of torus knots, which determines for some cases.

 Speaker： Ikuo Tayama (OCAMI, COE researcher) Title： Enumerating 3-manifolds with lengths up to 9 by a canonical order 　(joint work with Akio Kawauchi (Osaka City University))

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 list (with several pending manifolds) of the enumeration of $3-$manifolds with lengths up to 9 by using the enumeration of the prime link exteriors.

 Speaker： Daniel Moskovich Title： Two Surgery Presentations for Dihedral Covering Spaces (joint work with Andrew Kricker (Nanyang Technological University))

This is joint work with A. Kricker. We present two different but related procedures for obtaining surgery presentations of dihedral covering spaces of $S3$ branched along knots. In particular we show that for any odd prime p there are exactly p equivalence classes of p-coloured knots modulo surgery by unit-framed (unknotted) components representing elements in the kernel of the p-colouring.

 Speaker： Noboru Ito (Waseda University) Title： Invariants via word for curves

We construct an 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.

 Speaker： Yo'av Rieck (University of Arkansas) Title： On the Heegaard genus of knot exteriors (joint with Tsuyoshi Kobayashi (Nara Women's University))

We will survey some of the authors' results about the behavior of Heegaard genus of knot exteriors under connected sum operation. As our main result we will prove that given integers $g_i > 1 (i=1,...,n)$, there exist knots $K_i$ in $S3$ so that:

1) $g(E(K_i)) =g_i$, and:
2) $g(E(K_1\sharp ...\sharp K_n)) = g(E(K_1)) +...+ g(E(K_n))$.

This proves the existence of counterexamples to Morimoto's Conjecture.

 Speaker： Teruhisa Kadokami (OCAMI, COE researcher) Title： Calculating the Casson-Walker invariants of cyclic branched coverings over knots yielding lens spaces (a joint work with Yasuyoshi Tsutsumi (Oshima National College of Maritime Technology) and Yukihiro Tsutsumi (Sophia University))

We exhibit a technique for calculating concretely the Casson-Walker invariants of cyclic branched coverings over knots yielding lens spaces on special cases. By using it, we consider problems such as detecting problem of branched covering spaces, and lens surgery problem in a homology 3-spheres.

 Speaker： Yoshihiro Fukumoto (Tottori University of Environmental Studies) Title： Homology spin cobordism problem of plumbed 3-manifolds and the cup product structures

In this talk, we introduce two approaches to give necessary conditions for abstract isomorphisms on homology of two plumbed 3-manifolds to be realized by homology spin cobordisms geometrically. In the first approach we use a V-manifold version of the Furuta-Kametani 10/8-inequality for closed spin 4-manifolds to obtain a necessary condition in terms of an integral lift of the Rochlin invariant and the quadruple cup product structure. In particular, we formally calculate cup products by using the data of abstract isomorphism on homology between plumbed 3-manifolds. In the second approach we use the associativity of cup products. In fact, the formal calculations of cup products may fail to satisfy the associativity law. Motivated by this, we introduce a certain triple product to give other necessary conditions.

 Speaker： Takao Satoh (Department of Mathematics Graduate School of Science, Osaka University) Title： Twisted homology groups of the automorphism group of a free group knots

In this talk, we compute twisted first and second homology groups of the automorphism group of a free group with coefficients in the abelianization of a free group and its dual group, using a presentation sm group of a free group due to Gersten.

 Speaker： Toshifumi Tanaka (OCAMI, COE fellow) Title： Maximal Thurston-Bennequin numbers and Rasmussen invariants of doubled knots

Maximal Thurston-Bennequin number is a knot invariant from contact geometry. It is well-known that the invariant is strictly less than the minimum v-degree of the Kauffman polynomial in the framing variable v. Recently, we showed that the Kauffman bound is sharp for any positive knot and any alternating knot. However, it is known to be not sharp for many other knots in general. In this talk, we confirm that the Kauffman bound is sharp for any double of a knot if the bound is sharp for the knot. On the other hand, it is also known that the maximal Thurston-Bennequin number is strictly less than Rasmussen's s-invariant. We also discuss the s-invariant bound for the maximal Thurston-Bennequin numbers of doubled knots.

 Speaker： Ushijima Akira (Graduate School of Natural Science and Technology, Kanazawa University) Title： Hyperbolic spatial graphs coming from strongly invertible knots (joint work with Kazuhiro Ichihara (Nara University of Education))

There is a way, called "rational-fold cyclic branched covering," to construct spatial graphs from an invertible knot. In this talk we will give a condition, which is expected to be a necessary and sufficient one, for strongly invertible knots in the 3-sphere to yield hyperbolic spatial graphs. We will also see that strongly invertible simple knots and tunnel number one knots satisfy it so that we can have infinitely many hyperbolic spatial graphs in the 3-sphere.

 Speaker： Atsushi Ishii (Research Institute for Mathematical Sciences, Kyoto University) Title： The skein index for link invariants

We introduce the skein index, which is an integer-valued index t is used to compare link invariants and to find a skein relation. We give the complete list for link invariants of skein index less than or equal to two, and discuss the skein index of an operator invariant.

 Speaker： Shin Satoh (Department of Mathematics, Kobe University) Title： The sheet numbers of 2-knots

A $2$-knot is an embedded $2$-sphere in $4$-space, and its diagram is a projection image of the $2$-knot into $3$-space together with crossing information. Such a diagram is regarded as a disjoint union of compact connected surfaces each of which is called a sheet. The sheet number of a $2$-knot is defined as the minimal number of sheets for all possible diagrams of the $2$-knot. The notion of the sheet number is analogous to the crossing number of a classical knot in $3$-space. In this talk, we give a lower bound of the sheet number in several ways (Fox colorings, fundamental quandles, and cocycle invariants), and determine the sheet numbers of the $2$-, $3$-twist-spun trefoils, and some ribbon $2$-knots.

 Speaker： Iwakiri Masahide (OCAMI, COE fellow) Title： Quandle cocycle invariants of charts with six white vertices

By $S_p(\Gamma)$, we denote the quandle cocycle invariant of a chart $\Gamma$ associated with Mochizuki's $3$-cocycle of the dihedral quandle of order $p$. In this talk, we prove that there is no chart $\Gamma$ with six white vertices such that $S_p(\Gamma)$ is non-trivial for some odd prime integer $p$ but $3$. This gives lower bounds of the w-indices of some surface links. We also consider charts with six white vertices such that $S_3(\Gamma)$ is non-trivial.

 Speaker： Motoo Tange (Department of Mathematics Graduate School of Science, Osaka University) Title： On tight contact structure and lens surgery

We will introduce well-known constraints of knots in 3-sphere which yield lens spaces by a positive Dehn surgery, and such knots. Secondly, we will generalize the constraints to general homology spheres and show such knots in case of L-space homology sphere. Finally we will show that there exist homology spheres that never admit lens surgery by using the contact structures.

 Speaker： Makiko Ishiwata (Hirasawa) (OCAMI, COE researcher) Title： A classification of links up to 5-move equivalence

Montesinous and Nakanishi conjectured that 3-move and 4-move are unknotting operations around 1980. In 2002, Dabkowski and Przytycki proved that 3-move is not an unknotting operation via argument of Burnside group, but 4-move conjecture remained unsettled. In this talk, we consider 5-move equivalence relation, and classify rational links and links up to 9-crossings.

 Speaker： YAMAMOTO, Ryosuke (OCAMI, COE researcher) Title： Complexity of open book decompositions via arc complex (Joint work with Toshio Saito (Nara Women's University))

Based on Hempel's distance of a Heegaard splitting, we define a certain kind of complexity of an open book decomposition, called a translation distance, by using the arc complex of its fiber surface. We then show that an open book decomposition is of translation distance at most two if it is split into "simpler" open book decompositions and at most three if it admits a Stallings twist on it.

 Speaker： Kouichi Yasui (Osaka University) Title： Small exotic rational surfaces without 1- and 3-handles

Jongil Park \textit{et al.}~constructed exotic $\mathbf{CP}^2\# n\overline{\mathbf{CP}}^2\,(5\leq n\leq 8)$ by using rational blow-downs and elliptic fibrations. In this talk we give another construction by using rational blow-downs and Kirby calculus. We also prove our manifolds admit handle decompositions without $1$- and $3$-handles, in the case $7\leq n\leq 9$. Note that every exotic $\mathbf{CP}^2$, if it exists, has at least either a $1$- or $3$-handle in each handle decompositon of it.