The 21st Century COE Program
Constitution of wideangle mathematical basis focused on knots













Department of Mathematics and Physics
Graduate School of Science
Osaka City University





Organizer in 2007 : TANAKA, Toshifumi 


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

（Abstract）
（PDF） 
Date 
：March 28 (Fri.) 15：00～16：00 
Place 
：Dept. of Mathematics, Sci. Bldg., 3153 

Top 
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 

Top 
Speaker 
：Takuji Nakamura (Osaka ElectroCommunication 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 

Top 
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 

Top 
Speaker 
：Hiromasa Moriuchi (OCAMI, COE researcher) 
Title 
：Classifications of thetacurves and handcuff graphs 

（Abstract）
（PDF） 
Date 
：February 8 (Fri.) 16：00～17：00 
Place 
：Dept. of Mathematics, Sci. Bldg., 3153 

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

（Abstract）
（PDF） 
Date 
：February 1 (Fri.) 16：00～17：00 
Place 
：Dept. of Mathematics, Sci. Bldg., 3153 

Top 
Speaker 
：Taizo Kanenobu (Osaka City University) 
Title 
：The sharpunknotting number of a torus knot 

（Abstract）
（PDF） 
Date 
：December 14 (Fri.) 13：30～14：30 
Place 
：Dept. of Mathematics, Sci. Bldg., 3153 

Top 
Speaker 
：Ikuo Tayama (OCAMI, COE researcher) 
Title 
：Enumerating 3manifolds 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 

Top 
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 

Top 
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 

Top 
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 

Top 
Speaker 
：Teruhisa Kadokami (OCAMI, COE researcher) 
Title 
：Calculating the CassonWalker 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 

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

（Abstract）
（PDF） 
Date 
：October 12 (Fri.) 14：00～15：00 
Place 
：Dept. of Mathematics, Sci. Bldg., 3153 

Top 
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 

Top 
Speaker 
：Toshifumi Tanaka (OCAMI, COE fellow) 
Title 
：Maximal ThurstonBennequin numbers and Rasmussen invariants of doubled knots 

（Abstract）
（PDF） 
Date 
：July 6 (Fri.) 16：00～17：00 
Place 
：Dept. of Mathematics, Sci. Bldg., 3153 

Top 
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 

Top 
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 

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

（Abstract）
（PDF） 
Date 
：June 8 (Fri.) 16：00～17：00 
Place 
：Dept. of Mathematics, Sci. Bldg., 3153 

Top 
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 

Top 
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 

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

（Abstract）
（PDF） 
Date 
：April 27 (Fri.) 16：00～17：00 
Place 
：Dept. of Mathematics, Sci. Bldg., 3153 

Top 
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 

Top 

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

（Abstract）
（PDF） 
Date 
：April 13 (Fri.) 16：00～17：00 
Place 
：Dept. of Mathematics, Sci. Bldg., 3153 

Top



Speaker： 
Masahiro Kawami (OCAMI, COE Reseacher) 
Title： 
Mod4 quadratic forms and diffeomorphisms on nonorientable surfaces 
We investigate Mod4 quadratic forms on the $\mathbb{Z}_{2}$coefficient
first homology group of a nonorientable 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 selfdiffeomorphims on a nonorientable
surface which leave a given Mod4quadratic 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 selfhomeomorphism on the common target space of the
invariants.
Speaker： 
Takuji Nakamura (Osaka ElectroCommunication 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 righthanded 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 thetacurves and handcuff graphs 
We have enumerated all the thetacurves 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 2Bridge Knots 
In joint work with Maybrun and Robinson, we prove that for 2bridge 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 sharpunknotting number of a torus knot 
The sharpunknotting number was introduced by Hitoshi Murakami. He estimates
for the sharpunknotting number from below using the signature, Arf invariant,
homology invariant from a cyclic covering. We give an estimation for the
sharpunknotting numbers of torus knots, which determines for some cases.
Speaker： 
Ikuo Tayama (OCAMI, COE researcher) 
Title： 
Enumerating 3manifolds 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 wellorder was introduced on the set of links by A. Kawauchi. This wellorder also naturally induces a wellorder on the set of prime link exteriors and eventually induces a wellorder 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 pcoloured knots modulo surgery
by unitframed (unknotted) components representing elements in the kernel
of the pcolouring.
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 CassonWalker 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 CassonWalker 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 3spheres.
Speaker： 
Yoshihiro Fukumoto (Tottori University of Environmental Studies) 
Title： 
Homology spin cobordism problem of plumbed 3manifolds and the cup product
structures 
In this talk, we introduce two approaches to give necessary conditions
for abstract isomorphisms on homology of two plumbed 3manifolds to be
realized by homology spin cobordisms geometrically. In the first approach
we use a Vmanifold version of the FurutaKametani 10/8inequality for
closed spin 4manifolds 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 3manifolds. 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 ThurstonBennequin numbers and Rasmussen invariants of doubled
knots 
Maximal ThurstonBennequin number is a knot invariant from contact geometry.
It is wellknown that the invariant is strictly less than the minimum vdegree
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 ThurstonBennequin number is strictly less
than Rasmussen's sinvariant. We also discuss the sinvariant bound for
the maximal ThurstonBennequin 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 "rationalfold 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 3sphere 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 3sphere.
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 integervalued 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 2knots 
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 $2$, $3$twistspun 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 nontrivial for some odd prime integer
$p$ but $3$. This gives lower bounds of the windices of some surface links.
We also consider charts with six white vertices such that $S_3(\Gamma)$
is nontrivial.
Speaker： 
Motoo Tange
(Department of Mathematics Graduate School of Science, Osaka University) 
Title： 
On tight contact structure and lens surgery 
We will introduce wellknown constraints of knots in 3sphere 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 Lspace 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 5move equivalence 
Montesinous and Nakanishi conjectured that 3move and 4move are unknotting
operations around 1980. In 2002, Dabkowski and Przytycki proved that 3move
is not an unknotting operation via argument of Burnside group, but 4move
conjecture remained unsettled. In this talk, we consider 5move equivalence
relation, and classify rational links and links up to 9crossings.
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 3handles 
Jongil Park \textit{et al.}~constructed exotic $\mathbf{CP}^2\# n\overline{\mathbf{CP}}^2\,(5\leq
n\leq 8)$ by using rational blowdowns and elliptic fibrations. In this
talk we give another construction by using rational blowdowns 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.
Last Modified on March 21, 2008.
All Rights Reserved, Copyright (c) 20032005 Department of Mathematics, OCU 