Friday Seminar on Knot Theory（2004年度）
21世紀COEプログラム「結び目を焦点とする広角度の数学拠点の形成 」
(Constitution of wide-angle mathematical basis focused on knots)
の事業の一環として、通例金曜日午後4：00～5：00に大阪市立大学数学研究所（OCAMI） で
「Friday Seminar on Knot Theory」を開始することになりました．
| 日時 |
2005年2月4日（金）16：00～17：00 |
| 講演者（所属） |
Eonkyung Lee（Sejong University） |
| タイトル |
Introduction to Cryptography via Braid Groups |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
This talk introduces modern cryptology via
braid groups.
First, we see cryptographic schemes, especially designed
using braid groups.
In braid groups, the conjugacy problem is known as a
pretty hard problem to solve.
Based on variants of this problem, there have been
proposed key agreement protocols,
public-key encryption schemes, pseudorandom number
generator, pseudorandom synthesizer,
entity authentication schemes, and so on.
Second, we see cryptanalysis of these schemes in two
steps.
At the first step, we see attacks mounted on them.
Here, we notice that these attacks are different from the
prior attacks in finite abelian groups.
Usually, probability argument is used in many cases when
analyzing attacks.
However, it is not in braid-group-based cryptanalysis.
This motivates to need a kind of analysis tools.
Therefore, at the second step, we see such a tool and
how to concretely analyze attacks in braid groups by using
it. |
| 日時 |
2005年1月28日（金）16：00～17：00 |
| 講演者（所属） |
Sang Jin Lee（Konkuk University） |
| タイトル |
A new approach to find a reduction system of reducible braids |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
The $n$-braid group $B_n$ is the group of automorphisms of
the $n$-punctured disk that fix the boundary point-wise,
modulo isotopy relative to the boundary.
A braid is \emph{reducible} if it fixes an essential
simple closed curve
system in the punctured disk.
The \emph{reducibility problem} is to decide, given a
braid, whether or not it is reducible.
In the this talk, we will survey the approaches to
the reducibility problem and present new results to the
problem using the canonical reduction system and Garside
theory. |
| 日時 |
2005年1月21日（金）16：00～17：00 |
| 講演者（所属） |
Nafaa Chbili（東京工業大学） |
| タイトル |
Graph-Skein Modules of Three-Manifolds |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
Let ${\mathcal{R}}={\Bbb Z}[{A^{\pm 1}},
\delta^{-1}]$, where $\delta=-A^2-A^{-2}$.
Let $M$ be a three-manifold and let $\mathcal{G}$ be the
set of all isotopy classes of ribbon graphs embedded in
$M$. We define the Yamada skein module
of M as the quotient of the free module $\mathcal{R[G]}$
by the skein relations introduced by S. Yamada
to define the topological invariant of spatial graphs
known as the Yamada polynomial. We
compute this module for Handelbodies and explore its
relationship with the Kauffman
bracket skein module. |
| 日時 |
2005年1月14日（金）16：00～17：00 |
| 講演者（所属） |
秋吉 宏尚（大阪大学大学院理学研究科, 特任研究員） |
| タイトル |
Ford domains of punctured torus groups and an application
to deformation theory |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
Let $M$ be the manifold obtained from the
product of the once-punctured torus and the closed
interval by Dehn surgery along an essential simple closed
curve in a level surface. In this talk, we will discuss
the combinatorial structure
of the Ford domain of the Kleinian group obtained as the
image of the holonomy representation of a hyperbolic
structure on $M$. The key is that the
Kleinian group which we consider is an amalgamated free
product of two punctured torus
groups, and that the combinatorial structure of the Ford
domain of a punctured
torus group is studied in detail by T. Jorgensen. |
| 日時 |
2004年12月17日（金）16：00～17：00 |
| 講演者（所属） |
宮澤 治子（津田塾大学・OCAMI, COE研究所員） |
| タイトル |
C_n-moves and polynomial link invariants as Vassiliev invariants |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
It is known that many polynomial
invariants are Vassiliev invariants.
On the other hand, it is also known that a $C_n$-move,
which is a local
move defined by K.Habiro,
does not change values of Vassiliev invariants of order
less than $n$.
In this talk, we report some relations between a
$C_n$-moves and polynomial
link invariants
which are Vassiliev invariants of order $n$ or $n+1$. |
| 日時 |
2004年12月3日（金）16：00～17：00 |
| 講演者（所属） |
斎藤 敏夫（大阪大学大学院理学研究科） |
| タイトル |
The dual knots of doubly primitive knots |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
It is one of the unsolved problems to decide the knots in
the $3$-sphere which have non-trivial Dehn surgery
yielding a lens space. The concept of doubly primitive
knots is introduced by Berge, and he proved that any
doubly primitive knot admits Dehn surgery yielding a lens
space. It is conjectured that Berge's construction is
complete. Also, he proved that the dual knots (in lens
spaces) of doubly primitive knots are $(1,1)$-knots. This
implies that it is important to study $(1,1)$-knots with
Dehn surgery yielding the $3$-sphere as well as knots in
the $3$-sphere with Dehn surgery yielding lens spaces.
In this talk, we will give some results on $(1,1)$-knots
with Dehn surgery yielding the $3$-sphere. In particular,
we give a necessary and sufficient condition for such
$(1,1)$-knots to be hyperbolic. |
| 日時 |
2004年11月26日（金）16：00～17：00 |
| 講演者（所属） |
Gregor Masbaum（パリ第7大学ジュシユ数学研究所） |
| タイトル |
Integral lattices in TQFT |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
We will describe joint work with Pat
Gilmer where we find explicit bases for naturally defined
lattices in the vector spaces associated to surfaces by
the SO(3) TQFT at an odd prime. These lattices form an
"Integral TQFT" in an appropriate
sense. Some applications relating quantum invariants to
classical 3-manifold topology will be given. |
| 日時 |
2004年11月19日（金）16：00～17：00 |
| 講演者（所属） |
山本 亮介（大阪大学大学院理学研究科） |
| タイトル |
Contact 3-manifolds and supporting open-book decompositions |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
Thurston and Winkelnkemper showed that
every $3$-manifold $M$ has a contact structure,
by giving a construction of a contact form on $M$ from
an open-book decomposition $(M,F)$ with a fiber surface
$F$. We say that the contact strucure is {\itshape
supported\/} by the open-book decompotion.
Giroux showed that every contact structure on $M$ is
supported by some open-book decomposition.
In this talk we will review the construction of a contact
structure, and discuss a relation between a property of
the monodromy map for
$F$ and the tightness of the contact structure.
In particular we give a characterization of a set of
simple closed curves on $F$ which may be Legendrian in the
contact structure and talk about some application of this
result. |
| 日時 |
2004年11月5日（金）16：00～17：00 |
| 講演者（所属） |
金信 泰造（大阪市立大学大学院理学研究科） |
| タイトル |
Commutator subgroups of certain 2-braid virtual knots |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
Let $G_n$ be the group of 2-braid virtual
knots of type $(-n, n-1, 1)$, $n \ge 2$,
which has $3$ virtual crossings,
and $G'_n$ its commutator subgroup.
Then we show:
\begin{equation*} G'_n \cong \begin{cases} {\bold Z}_2^n & \text{if
$n \equiv 0$, $1 \pmod{3}$;} \\ Q \times {\bold Z}_2^{n-2} & \text{if
$n \equiv 2 \pmod{3}$,} \end{cases} \end{equation*} where $Q$ is the quarternion
group of order $8$. In particular, the abelianized group of $G'_n$, $G'_n/G''_n$
is ${\bold Z}_2^n$. According to Shin Satoh, the group of a virtual knot
is that of a torus $2$- knot in $4$-sphere. Also, $G_2$ is the group of
the $3$-twist spun trefoil knot. However, $G_n$ with $n\ge 3$ is not a
$2$-knot group. In fact, Hillman has given all possible finite groups that
are the commutator subgroups of $2$-knot groups, and $G'_n$ with $n\ge
3$ is not contained in his list. |
| 日時 |
2004年10月29日（金）16：00～17：00 |
| 講演者（所属） |
浅田 雅彦（大阪市立大学大学院理学研究科） |
| タイトル |
On ch-diagrams with double points representing immersed surfaces
in 4-space |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
Any surface embedded in 4-space can be
represented by a certain 4-valent
plane graph having two kinds of vertices, which is called
a ch-diagram.
Using this diagram, K. Yoshikawa gave a method for
enumerating all
non-splittable and weakly prime embedded surfaces,
and a table of such surfaces represented by diagrams the
numbers of whose
vertices are up to ten.
S. Kamada indicated that any surface immersed in 4-space
having only
transverse double points as its singularities can be
represented
by a certain 4-valent plane graph with three kinds of
vertices,
which is called a ch-diagram with double points.
The speaker gave an analogous method to Yoshikawa's one
for enumerating
all non-splittable and weakly prime such immersed surfaces
by ch-diagrams with double points, and a table of all such
surfaces
represented by such diagrams with up to five vertices
and several surfaces done by ones with six vertices in his
dissertation.
Unfortunately, the speaker however found one omission in
the table (there
may exist the other omissions),
which is the knot sum of a standard projective plane and a
standard
immersed sphere.
What to be noticed is that this surface is non-prime but
weakly prime.
(The index of a diagram is the number of its vertices,
and that of a surface is the minimal number of indices
among all diagrams
representing the surface.
A surface $F$ is weakly prime if it can not be the knot
sum of any two
surfaces $F_1, F_2$ such that each index of them is less
than that of $F$.)
The speaker is now reconstructing such a table.
In this talk, he will review the method for enumerating
such surfaces, and
report the omission mentioned above. |
| 日時 |
2004年10月22日（金）16：00～17：00 |
| 講演者（所属） |
河内 明夫（大阪市立大学） |
| タイトル |
Quasi-torus links and distance by zero-linking twists |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
Given an oriented link $L$ and a trivial
knot $k$ in the 3-sphere with the linking number
Link$(L,k)=0$, we can obtain a link $L'$
from $L$ by twisting $L$ along $k$.
The operation $L\to L'$ is called a {\it zero-linking
twist}. Any two oriented links with the same number
of components are transformed each other by some number of
zero-linking twists.
In this talk, we first review an algebraic estimation
(given in Kobe J. Math. 13(1996),183-190) on the minimal
number of zero-linking twists needed to transform between
two given oriented links with the same number of
components.
By using this result, we estimate the distance
between a quasi-torus link of type $(p,q)$ introduced by
V. O. Munturov and the torus link of type $(p,q)$. This
result will be included in a
joint work with Yongju Bae and Seogman Seo. |
| 日時 |
2004年10月15日（金）16：00～17：00 |
| 講演者（所属） |
田山 育男（大阪市立大学大学院理学研究科） |
| タイトル |
Enumerating prime links by a canonical order |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
This work is a joint work with A.
Kawauchi. A well-order (called a {\it canonical order})
was introduced in the set of (unoriented) links by A.
Kawauchi [K] (see also A. Kawauchi and I. Tayama [KT]).
This well-order also naturally induces a well-order in the
set of closed connected orientable $3$-manifolds and
suggests a method for enumerating
the prime links and the $3$-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 and we call the 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 $L'$, and for any natural number $n$ there
are only finitely many, uniquely ordered links with
lengths up to $n$.
In this talk, we give a way to enumerate the prime links
by the canonical order and show a table of the prime links
with lengths up to 10. Our argument enables us to discover
7 omissions and one overlap
in Conway's table of links of 10 crossings.
References
[K]
A. Kawauchi, A tabulation of 3-manifolds via Dehn
surgery,
Boletin de la Sociedad Matematica Mexicana (to appear).
[KT] A. Kawauchi and I. Tayama,
Enumerating the prime knots and links by a canonical
order,
in: Proc. First East Asian School of Knots, Links, and
Related Topics
(Seoul, Feb. 2004), (2004)307-316.
(Online Version) http://knot.kaist.ac.kr/2004/proceedings.php
|
| 日時 |
2004年10月8日（金）16：00～17：00 |
| 講演者（所属） |
堤 康嘉（大阪市立大学大学院理学研究科） |
| タイトル |
The Casson-Walker-Lescop invariant for branched cyclic
covers of $S^{3}$ branched over a 2-bridge knot |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
Let $D(b_{1},\ldots,b_{2m})$ be a 2-bridge knot, let
$M^{r}_{b_{1}, \ldots
,b_{2m}}$ be
the $r$-fold cyclic covering of $S^3$ branched over
$D(b_{1},\ldots,b_{2m})$.
We compute the Casson-Walker-Lescop invariant of
$M^{r}_{b_{1}, \ldots
,b_{2m}}$ by
using Lescop's formula and Chbili's method. |
| 日時 |
2004年10月1日（金）16：00～17：00 |
| 講演者（所属） |
中村 拓司（OCAMI） |
| タイトル |
On the minimal genus of knots via braidzel surafces |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
A notion of braidzel surfaces has
introduced by L. Rudolph as a generalization of pretzel
surfaces on his study of the quasipositivity for pretzel
surfaces. The speaker showed that any knot
bounds an orientable braidzel surface.
By this fact, we define the ``genus'' for knots with
respect to their braidzel surfaces. The minimal genus
among all oriented braidzel surfaces for a knot $K$ is
defined to be the {\it braidzel genus} for $K$, denoted by
$g_b(K)$. In this talk, we discuss relationships among
the braidzel genus and other `genus' for knots. |
| 日時 |
2004年9月24日（金）16：00～17：00 |
| 講演者（所属） |
門上 晃久（OCAMI） |
| タイトル |
Surface bracket polynomial and supporting genus of virtual knots |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
Non-triviality of Kishino's (virtual)
knot cannot be proved by the fundamental group, the
Kauffman bracket polynomial (the Jones polynomial) and the
Sawollek polynomial. T. Kishino proved its non-triviality
by the 3-strand bracket polynomial.
T. Kadokami showed that the supporting genus of flat
Kishino's knot is two.
This means Kishino's knot is non-classical. H. A. Dye and
L. Kauffman defined the
{\it surface bracket polynomial} for virtual links. We
talk about this invariant.
The invariant detects non-classicality of many virtual
knots including Kishino's knot
whose bracket polynomials are trivial, and the supporting
genus of some virtual knot
can be determined by the invariant.
References:
H. A. Dye and L. H. Kauffman,
Minimal surface representations of virtual knots and
links,
math.GT/0401035. |
| 日時 |
2004年6月25日（金）16：00～17：00 |
| 講演者（所属） |
Seo Seogman（Kyungpook National University） |
| タイトル |
QUASITORIC BRAIDS OF LINKS |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
We will introduce new link invariants which are called
the cycle length and the quasitoric braid index of link
and show that the braid index of knots and the quasitoric braid
index of knots are the same, by using the cycle length of knots. |
| 日時 |
2004年6月18日（金）16：00～17：00 |
| 講演者（所属） |
Yo'av Rieck（University of Arkansas） |
| タイトル |
Recent results about thin position |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
Thin position was introduced by Gabai to study
Property R, and
was an important ingredient in several studies, for
example Gordon--Luecke's
proof that knots are determined by their complements. In
recent years thin
position has appeared as the topic of several works, where
different
authors tried to understand the behavior of knots in thin
position. In this talk
we will first define thin position and then discuss some
of these results,
of (among others) Heath-Kobayashi, Hendricks,
Rieck-Sedgwick,
Scharlemann-Schultens and Scharlemann-Thompson. We intend
for this talk to be an
expository talk and hence appropriate for graduate
students. |
| 日時 |
2004年6月11日（金）16：00～17：00 |
| 講演者（所属） |
Alexander Stoimenow（University of Toronto） |
| タイトル |
On mutations and Vassiliev invariants (not) contained
in knot polynomials |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
It is known that the
Brandt-Lickorish-Millett-Ho polynomial $Q$
contains Casson's knot invariant. Whether there are
(essentially)
other Vassiliev knot invariants obtainable from $Q$ is an
open
problem. We show that this is not so up to degree $\le 9$.
We also
give the (apparently) first example of knots not
distinguished by
2-cable HOMFLY polynomials, which are not mutants. Our
calculations
provide evidence against the conjecture that Vassiliev
knot invariants
of degree $\le 10$ are determined by the HOMFLY and
Kauffman polynomial
and their 2-cables, and for the existence of algebras of
such Vassiliev
invariants not isomorphic to the algebras of their weight
systems. |
| 日時 |
2004年6月4日（金）16：00～17：00 |
| 講演者（所属） |
屋代 司（大阪市立大学） |
| タイトル |
Crossing changes and attaching handles for surface--knots |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
A surface--knot is an embedded connected
orientable surface in 4--space.
A surface diagram is a generic projection of a
surface--knot into 3--space
with crossing information.
A surface diagram may contain double curves, triple points
and
branch points.
Some surface diagrams have special closed double curves,
in which
we can apply crossing changes to obtain a trivial surface.
On the other hand, attaching some 1--handles to a
surface--knot,
we can obtain a trivial surface.
In this talk we will show that a crossing change
on a surface diagram relates to attaching 1--handles to
the surface diagram.
We will demonstrate those operations by a sequence of
local deformations
on surface diagrams. |
| 日時 |
2004年5月28日（金）16：00～17：00 |
| 講演者（所属） |
Rama Mishra（Department of Mathematics Indian Institute of Technology） |
| タイトル |
Polynomial Knots and their Degree Sequence |
| 場所 |
数学 第3セミナー室（3153） |
| アブストラクト |
Polynomial Knot is a smooth embedding
of $R$ in $R^3$ defined by
$t\mapsto(f(t), g(t), h(t))$ where $f(t)$, $g(t)$ and
$h(t)$ are polynomials over
the field of real numbers. They represent {\it non-compact
knots} (introduced
by Vassiliev). Two polynomial knots are said to be
equivalent
if there exists a one parameter family of polynomial
embeddings of
$R$ in $R^3$ connecting one to the other. It has been
proved that every
non-compact knot is ambient isototopic to a polynomial
knot. If a
polynomial knot is given by an embedding $t\mapsto(f(t),
g(t), h(t))$ where
$\deg f(t) = l$, $\deg(g(t)) = m$ and $\deg(h(t)) = n$
then we say that
$(l, m, n)$ is a degree sequence of the knot $K$. Degree
sequence for a
given knot may not be unique. If a degree sequence $(l, m,
n)$ for a
given knot is minimal in the sense of lexicographic
ordering of $N^3$
then it is called the {\it minimal degree sequence} of
that knot. In this
talk we shall discuss degree sequences of knots and
minimal degree
sequence for some important class of knots. |
最終更新日: 2005年4月5日
