Friday Seminar on Knot Theory (2004)
| Date | Feb. 4 (Fri.) 16：00～17：00 |
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| Speaker | Eonkyung Lee（Sejong University） |
| Title | Introduction to Cryptography via Braid Groups |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Jan. 28 (Fri.) 16：00～17：00 |
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| Speaker | Sang Jin Lee（Konkuk University） |
| Title | A new approach to find a reduction system of reducible braids |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Jan. 21 (Fri.) 16：00～17：00 |
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| Speaker | Nafaa Chbili (Tokyo Institute of Technology) |
| Title | Graph-Skein Modules of Three-Manifolds |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Jan. 14 (Fri.) 16：00～17：00 |
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| Speaker | Hirotaka Akiyoshi（Osaka University） |
| Title | Ford domains of punctured torus groups and an application to deformation theory |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Dec. 17 (Fri.) 16：00～17：00 |
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| Speaker | Haruko Miyazawa（Tsuda College・OCAMI, COE Research Member） |
| Title | C_n-moves and polynomial link invariants as Vassiliev invariants |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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 $n+1$. |
| Date | Dec. 3（Fri.） 16：00～17：00 |
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| Speaker | Toshio Saito (Osaka University) |
| Title | The dual knots of doubly primitive knots |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Nov. 26（Fri.） 16：00～17：00 |
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| Speaker | Gregor Masbaum（Institut de Mathematiques de Jussieu） |
| Title | Integral lattices in TQFT |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Nov. 19（Fri.） 16：00～17：00 |
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| Speaker | Ryosuke Yamamoto (Osaka University) |
| Title | Contact 3-manifolds and supporting open-book decompositions |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Nov. 5（Fri.） 16：00～17：00 |
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| Speaker | Taizo Kanenobu (Osaka City University) |
| Title | Commutator subgroups of certain 2-braid virtual knots |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Oct. 29（Fri.） 16：00～17：00 |
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| Speaker | Masahiko Asada (Osaka City University) |
| Title | On ch-diagrams with double points representing immersed surfaces in 4-space |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Oct. 22（Fri.） 16：00～17：00 |
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| Speaker | Akio Kawauchi (Osaka City University) |
| Title | Quasi-torus links and distance by zero-linking twists |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Oct. 15（Fri.） 16：00～17：00 |
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| Speaker | Ikuo Tayama (Osaka City University) |
| Title | Enumerating prime links by a canonical order |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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 |
| Date | Oct. 8（Fri.） 16：00～17：00 |
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| Speaker | Yasuyoshi Tsutsumi (Osaka City University) |
| Title | The Casson-Walker-Lescop invariant for branched cyclic covers of $S^{3}$ branched over a 2-bridge knot |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Oct. 1（Fri.） 16：00～17：00 |
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| Speaker | Takuji Nakamura（OCAMI） |
| Title | On the minimal genus of knots via braidzel surafces |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Sep. 24（Fri.） 16：00～17：00 |
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| Speaker | Teruhisa Kadokami（OCAMI） |
| Title | Surface bracket polynomial and supporting genus of virtual knots |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Jun. 25（Fri.） 16：00～17：00 |
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| Speaker | Seo Seogman（Kyungpook National University） |
| Title | QUASITORIC BRAIDS OF LINKS |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Jun. 18（Fri.） 16：00～17：00 |
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| Speaker | Yo'av Rieck（University of Arkansas） |
| Title | Recent results about thin position |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Jun. 11（Fri.） 16：00～17：00 |
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| Speaker | Alexander Stoimenow（University of Toronto） |
| Title | On mutations and Vassiliev invariants (not) contained in knot polynomials |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | Jun. 4（Fri.） 16：00～17：00 |
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| Speaker | Tsukasa Yashiro (Osaka City University） |
| Title | Crossing changes and attaching handles for surface--knots |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |
| Date | May 28（Fri.） 16：00～17：00 |
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| Speaker | Rama Mishra（Department of Mathematics Indian Institute of Technology） |
| Title | Polynomial Knots and their Degree Sequence |
| Place | Dept. of Mathematics, Sci. Bldg., 3153 |
| Abstract | 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. |