市大数学教室

大阪市立大学大学院理学研究科数物系専攻 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)
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  Friday Seminar on Knot Theory(2005年度)
2004年度へ  2006年度へ


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.


講 演 者 :栗屋隆仁 (九州大学大学院数理学府・日本学術振 興会特別研究員)
タ イ ト ル :Remark on a spin refinement of the perturbative invariant
 (アブストラクト)    (PDF
日 時 :1月20日(金)16:10~17:10
場 所 :数学 第3セミナー室(3153)
Toptop
講 演 者 :谷口 太聖(慶應義塾大学)
タ イ ト ル :Turaev-Viro invariant for all orientable Seifert manifolds
 (アブストラクト)    (PDF
日 時 :12月16日(金)16:10~17:10
場 所 :数学 第3セミナー室(3153)
Toptop
講 演 者 :古宇田 悠哉
(慶應義塾大学大学院理工学研究科,日本学術振興会特別研究員)
タ イ ト ル :Heegaard-type presentations of branched standard spines and Reidemeister-Turaev torsion
 (アブストラクト)    (PDF
日 時 :12月9日(金)16:10~17:10
場 所 :数学 第3セミナー室(3153)
Toptop
講 演 者 :Rama Mishra (JSPS Researcher, Osaka City University,
                      Indian Institute of technology-Delhi, India)
タ イ ト ル :In search of an ideal polynomial representation of a knot-type
 (アブストラクト)    (PDF
日 時 :12月9日(金)15:00~16:00
場 所 :数学 第3セミナー室(3153)
Toptop
講 演 者 :金信 泰造(大阪市立大学大学院理学研究科・数学教室)
タ イ ト ル :Skein Relation for the HOMFLYPT Polynomials of Two-Cable Links
 (アブストラクト)    (PDF
日 時 :12月2日(金)16:00~17:00
場 所 :数学 第3セミナー室(3153)
Toptop
講 演 者 :堤 幸博(上智大学理工学部,日本学術振興会特別研究員)
タ イ ト ル :On the equivariant Casson invariant for fibered knots
 (アブストラクト)    (PDF
日 時 :11月25日(金)16:00~17:00
場 所 :数学 第3セミナー室(3153)
Toptop
講 演 者 :金 英子 (京都大学 リサーチフェロー)
タ イ ト ル :A search of braids which have smaller dilatation than given pseudo-Anosov braids
 (アブストラクト)    (PDF
日 時 :11月18日(金)13:30~14:30
場 所 :数学 第3セミナー室(3153)
Toptop
講 演 者 :Nafaa Chbili (大阪市立大学数学研究所、COE Research member)
タ イ ト ル :Symmetry in Dimension three: A quantum approach
 (アブストラクト)    (PDF
日 時 :11月4日(金)16:00~17:00
場 所 :数学 第3セミナー室(3153)
Toptop
講 演 者 :川村 友美(青山学院大学理工学部物理・数理学科)
タ イ ト ル :The Rasmussen invariants and the sharper slice-Bennequin inequality on knots
 (アブストラクト)    (PDF
日 時 :10月28日(金)16:00~17:00
場 所 :数学 第3セミナー室(3153)
Toptop
講 演 者 :田中 心(東京大学大学院数理科学研究科・日本学術振興会特別研究員)
タ イ ト ル :Inequivalent surface-knots with the same knot quandle
 (アブストラクト)    (PDF
日 時 :10月21日(金)16:00~17:00
場 所 :数学 第3セミナー室(3153)
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講 演 者 :児玉 宏児(神戸高専)
タ イ ト ル :Computing technic of knot theory
 (アブストラクト)    (PDF
日 時 :10月7日(金)16:00~17:00
場 所 :数学 第3セミナー室(3153)
Toptop
講 演 者 :鎌田 直子(大阪市立大学)
タ イ ト ル :An algorithm to calculate Miyazawa polynomials of virtual knots
 (アブストラクト)    (PDF
日 時 :9月30日(金)16:00~17:00
場 所 :数学 第3セミナー室(3153)
Toptop
講 演 者 Fengchun Lei (Harbin Institute of Techonology/ハルビン工業大学)
タ イ ト ル :On Maximal Collections of Essential Annuli in a Handlebody
  (アブストラクト)    (PDF
日 時 :9月 2日(金) 16:00~17:00
場 所 :数学 第3セミナー室(3153)
Toptop
講 演 者 :山本 稔(北海道大学大学院理学研究科)
タ イ ト ル :The minimal number of singular set for fold maps
  (アブストラクト)    (PDF
日 時 :7月 15日(金) 16:00~17:00
場 所 :数学 第3セミナー室(3153)
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講 演 者 :屋代 司 (学振PD・大阪市立大学大学院理学研究科)
タ イ ト ル :Cross-exchangeable cycles and 1-handles for surface diagrams
  (アブストラクト)    (PDF
日 時 :7月 8日(金) 16:00~17:00
場 所 :数学 第3セミナー室(3153)
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講 演 者 :Tom Fleming(University of California, San Diego (UCSD))
タ イ ト ル :Chirality of Alternating Knots in S x I
  (アブストラクト)    (PDF
日 時 :7月 1日(金) 17:00~18:00
場 所 :数学 第3セミナー室(3153)
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講 演 者 :田中 利史(東京大学大学院数理科学研究科,COE研究員)
タ イ ト ル :Ribbon 2-knots assoiciated with symmetric unions
  (アブストラクト)    (PDF
日 時 :7月 1日(金) 16:00~17:00
場 所 :数学 第3セミナー室(3153)
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講 演 者 :田山 育男(大阪市立大学数学研究所)
タ イ ト ル :Enumerating the exteriors of prime links by a canonical order
  (アブストラクト)    (PDF
日 時 :6月 24日(金) 16:00~17:00
場 所 :数学 第3セミナー室(3153)
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講 演 者 :鈴木 正明(東京大学数理)
タ イ ト ル :Twisted Alexander invariant and a partial order in the knot table
  (アブストラクト)    (PDF
日 時 :6月 10日(金) 16:00~17:00
場 所 :数学 第3セミナー室(3153)
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講 演 者 :市原 一裕(大阪産業大学教養部)
タ イ ト ル :Alexander polynomials of doubly primitive knots
    (joint work with Toshio Saito (Osaka University) 
   and Masakazu Teragaito (Hiroshima University))
  (アブストラクト)    (PDF
日 時 :6月 3日(金) 16:00~17:00
場 所 :数学 第3セミナー室(3153)
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講 演 者 :佐藤 進(千葉大学大学院自然科学研究科)
タ イ ト ル :Disk presentations of surface-knots and -links
  (アブストラクト)    (PDF
日 時 :5月 20日(金) 16:00~17:00
場 所 :数学 第3セミナー室(3153)
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講 演 者 :中西 清孝(九州大学大学院数理学府)
タ イ ト ル :Sliding relation in the Kauffman bracket skein module of a 2-bridge knot exterior
  (アブストラクト)    (PDF
日 時 :5月 13日(金) 16:00~17:00
場 所 :数学 第3セミナー室(3153)
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講 演 者 :石井 敦(大阪大学大学院理学研究科)
タ イ ト ル :Skein relations for the generalized Alexander polynomial for virtual
    links and closed virtual 2-braids
  (アブストラクト)    (PDF
日 時 :5月 6日(金) 16:00~17:00
場 所 :数学 第3セミナー室(3153)
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アブストラクト集



講 演 者: 栗屋隆仁 (九州大学大学院数理学府・日本学術振興会特別研究員)
タ イ ト ル: Remark on a spin refinement of the perturbative invariant

In this talk, we will give a brief review of a spin refinement of the perturbative invariant defined by A.Beliakova, C.Blanchet and T.Le, and we will try to construct a invariant of spin 3-manifolds.
When the time allows, we talk on some related topics.

Toptop



講 演 者: 谷口 太聖(慶應義塾大学)
タ イ ト ル: Turaev-Viro invariant for all orientable Seifert manifolds

Turaev-Viro invariant is a topological invariant for closed 3-manifolds. In this talk, we introduce the definition of TV-invariant by using special spines of closed 3-manifolds, and give a formula of TV-invariant for all orientable Seifert manifolds. Our formula is based on a new construction of special spines of all orientable Seifert manifolds and the "gluing lemma" of topological quantum field theory. By using this formula, we get sufficient conditions that TV-invariant of Seifert manifolds coincides and make a computer program calculating TV-invariant of orientable Seifert manifolds. The method of constructing our formula is applicable to the "state sum" type invariant, for example Turaev-Viro-Ocneanu invariant or Dijkgraaf-Witten invariant.

Toptop



講 演 者: 古宇田 悠哉
(慶應義塾大学大学院理工学研究科,日本学術振興会特別研究員)
タ イ ト ル: Heegaard-type presentations of branched standard spines and Reidemeister-Turaev torsion

Reidemeister-Turaev torsion is an invariant of a 3-manifold $M$ equipped with a $\mathrm{Spin}^c$ structure, one representation of which is a homology class of a non-singular vector field on $M$. In this talk, we introduce a way to represent a branched standard spine, which can be regarded as a combinatorial presentation of a $\mathrm{Spin}^c$ structure on a 3-manifold, as a Heegaard diagrams with "mark", and explain an accessible way to compute the invariant using this presentation. We also explain some interesting behavior of this invariant by using examples, and relation to Seiberg-Witten invariant.

Toptop



講 演 者: Rama Mishra (JSPS Researcher, Osaka City University,
                      Indian Institute of technology-Delhi, India)
タ イ ト ル: In search of an ideal polynomial representation of a knot-type

One of the central themes of {\it geometric knot theory} is to find an ideal configuration of knot-types in a specified category. Here we consider the space $\mathbb{P}$ of all polynomial knots. Following Vassiliev's notation $\mathbb{P}=K3\setminus \Sigma,$ where $K3$ is the set of all maps $\phi:\mathbb{R}\longrightarrow\mathbb{R}^3$ defined by $\phi(t)=(x(t),y(t),z(t))$ where $x(t)=t^d+a_1t^{d-1} +\cdots+a_{d-1}t$, $y(t)= t^d+b_1t^{d-1}+\cdots+b_{d-1}t,$ and $z(t) =t^d+c_1t^{d-1}+\cdots+c_{d-1}t)$ and $\Sigma$ is the descriminent space of $K3.$ The topology of the space $\mathbb{P}$ is still under investigation. It is proved that any $C1$ knot in $S3$ is isotopy equivalent to the closure of the image of such a map for some degree $d$. Also the path components of $\mathbb{P}$ correspond to a knot-type. It can be easily proved that nontrivial knots cannot be realized by such maps of degree less than 5. As, polynomial represenatation for a given knot-type is not unique, the question of choosing an ideal poynomial representation makes sense. We have made an effort to define an energy function on the space $\mathbb{P}$ and based on this function we call a polynomial representation of a given knot-type with minimum energy to be the ideal one. We shall discuss if such an ideal representation for a given knot-type exists?

Toptop



講 演 者: 金信 泰造(大阪市立大学大学院理学研究科・数学教室)
タ イ ト ル: Skein Relation for the HOMFLYPT Polynomials of Two-Cable Links

We give a skein relation for the HOMFLYPT polynomials of 2-cable links. In [A. Ishii and T. Kanenobu, Different links with the same Links-Gould invariant, Osaka J. Math. 42 (2005) 273--290], we construct examples of arbitrarily many 2-bridge knots sharing the same HOMFLYPT, Kauffman, and Links-Gould polynomials, and arbitrarily many 2-bridge links sharing the same HOMFLYPT, Kauffman, Links-Gould, and 2-variable Alexander polynomials. Using the skein relation, we show their 2-cables also share the same HOMFLYPT polynomials.

Toptop



講 演 者: 堤 幸博(上智大学理工学部,日本学術振興会特別研究員)
タ イ ト ル: On the equivariant Casson invariant for fibered knots

Let $K$ be a fibered knot in $S3$ with ${\rm vol}(S3-K)=0$. Then $K$ is a fibered knot such that the monodromy is decomposed into periodic maps and the JSJ-family of the complement consists of Seifert fibered spaces. A theorem of O. Collin and N. Saveliev implies that for such a fibered knot $K$, the Casson invariant $\lambda(\Sigma^r_K)$ of the $r$-fold cyclic cover $\Sigma^r_K$ of $S3$ with branch set $K$ is written in terms of the equivariant knot-signatures of $K$ which are determined by the matrix of the monodromy (when $\Sigma^r_K$ is an integral homology sphere.) This is specific to fibered knots with ${\rm vol}(S3-K) = 0$ and is due to some property of the $SU(2)$-representation of $\pi_1(\Sigma^r_K)$ under the cyclic action on $\Sigma^r_K$.

Given a fibered knot with certain properties, one can construct infinitely many fibered knots with the same Seifert form and with distinct values of $\lambda(\Sigma^r_K)$ by adding a Dehn twist to the monodromy. We study the variation of $\lambda(\Sigma^r_K)$ under this construction and ${\rm vol}(S3-K)$ for fibered knots and non-fibered knots with trivial Alexander polynomial.

Toptop



講 演 者: 金 英子 (京都大学 リサーチフェロー)
タ イ ト ル: A search of braids which have smaller dilatation than given pseudo-Anosov braids

The dilatation is an invariant of pseudo-Anosov braids. Fixing strands n, the minimal of dilatation exists among pseudo-Anosov n-braids. Toward the determination of braids with minimal dilatation, we discuss how to find braids which have smaller dilatation than given pseudo-Anosov braids. A result by Los tells us that a given pseudo-Anosov braid $\beta$, any braid $\alpha$ dynamically forced by $\beta$ have always smaller dilatation than $\beta$. Such forced braid $\alpha$ can be captured by using the train track map associated to $\beta$. A problem is to know whether $\alpha$ is pseudo-Anosov or not, and how to compute the train track map associated to $\alpha$ when it is pseudo-Anosov. This is because in general, one can not predict how does the train track map of $\alpha$ look like. We give a solution of the problem when the train track map of $\beta$ contains "the star shaped rotational map".

Toptop



講 演 者: Nafaa Chbili (大阪市立大学数学研究所,COE Research member)
タ イ ト ル: Symmetry in Dimension three: A quantum approach

Let $M$ be an oriented compact three-manifold and $G$ a finite cyclic group of prime order. The manifold $M$ is said to be symmetric if $G$ acts non trivially on $M$. Several classical methods have been used to study the symmetries of three-manifolds. In this talk, we shall explain how to use the quantum invariants to study this problem. Namely, we will show how Murasugi's results on periodic knots have been extended to quantum invariants of three-manifolds by Chbili and Gilmer in the case of the $Su(2)$ and the $Su(3)$ quantum invariants. Then, by Chen-Le to allcomplex simple Lie algebras and by Qazaqzeh to modular categories.

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講 演 者: 川村 友美(青山学院大学理工学部物理・数理学科)
タ イ ト ル: The Rasmussen invariants and the sharper slice-Bennequin inequality on knots

Rasmussen introduced a knot invariant based on Khovanov homology theory, and showed that this invariant estimates the four-genus. We compare his result with the sharper slice-Bennequin inequality for knots. Then we obtain a similar estimate of the Rasmussen invariant to this inequality.
It includes the Bennequin inequality on the Rasmussen invariants, which is shown independently by Plamenevskaya and Shumakovitch.

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講 演 者: 田中 心(東京大学大学院数理科学研究科・日本学術振興会特別研究員)
タ イ ト ル: Inequivalent surface-knots with the same knot quandle

We have a knot quandle $Q(k)$ and a fundamental class $[k]$ as invariants for a classical knot $k$.
Similarly, we have a knot quandle $Q(F)$ and a fundamental class $[F]$ as invariants for a surface-knot $F$.

For classical knots, Joyce and Matveev independently proved that $Q(k)$ characterizes the classical knot $k$ up to reflected inverse,and Eisermann proved that the pair $Q(k)$ and $[k]$ characterize the classical knot $k$ completely.

We consider the following "hierarchy" for surface-knots $F$ and $F'$.
(i) There exists a quandle isomorphism $\phi :Q(F) \rightarrow Q(F')$.
(ii) There exists a quandle isomorphism $\phi :Q(F) \rightarrow Q(F')$ such that $\phi_{\ast}[F] = [F']$.
(ii)' There exists a quandle isomorphism $\phi :Q(F) \rightarrow Q(F')$ such that $\phi_{\ast}[F] = \pm [F']$.
(iii) The surface-knot $F$ is equivalent to $F'$.
(iii)' The surface-knot $F$ is equivalent to $F'$ or $-(F')^{\ast}$.

We note that (iii) $\Rightarrow$ (ii) $\Rightarrow$ (i) and (iii)' $\Rightarrow$ (ii)' $\Rightarrow$ (i) by definition.

In this talk, we illustrate the gap between (i) and (ii)', the gap between (ii)' and (iii)', and the gap between (ii) and (iii)
for surface-knots.

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講 演 者: 児玉 宏児(神戸高専)
タ イ ト ル: Computing technic of knot theory

"KNOT" is a computing tool on knot theory. In this talk, we will study how to implement or write topological idea as programming codes. Let us walk through source codes about skein invariants in "KNOT" program.

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講 演 者: 鎌田 直子(大阪市立大学)
タ イ ト ル: An algorithm to calculate Miyazawa polynomials of virtual knots

In 2004, Y. Miyazawa discovered a method to define polynomial invariants for a virtual knot, which are generalization of Kauffman's f-polynomial.
They enable us to distinguish Kishino's knot from a trivial knot. This August we announced a table of virtual knots with four real crossings classified by use of Miyazawa polynomials, JKSS invariants and 2-cabled Jones polynomials. In order to get the table and calculate invariants, we made a computer program. In this talk, we introduce an algorithm to calculate two kinds of Miyazawa polynomials from a Gauss chord diagram of a virtual knot.

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講 演 者: Fengchun Lei (Harbin Institute of Techonology/ハルビン工業大学)
タ イ ト ル: On Maximal Collections of Essential Annuli in a Handlebody

It is known that, for a maximal collection $\cal A$ of pairwise disjoint non-parallel essential annuli in a handlebody of genus 2, $1\leq|{\cal A}|\leq 3$. We show that for a maximal collection $\cal A$ of pairwise disjoint non-parallel essential annuli in a handlebody of genus $n$ $(\geq 3)$, $|{\cal A}|\leq 4n-5$, and the bound is best possible.

This is a joint work with Jingyan Tang.

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講 演 者: 山本 稔(北海道大学大学院理学研究科)
タ イ ト ル: The minimal number of singular set for fold maps

In 1970's, Eliashberg showed that every smooth map between oriented surfaces is homotopic to a fold map. A fold map is a smooth map which has only fold singularity ($A_1$-type singularity). It is known that for a fold map between surfaces, singular set is a one-dimensional submanifold in the source surface. In this talk, we will determine the minimal number of components of singular set for fold maps when we fix the mapping degree and the genuses of source and target oriented closed surfaces.

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講 演 者: 屋代 司 (学振PD・大阪市立大学大学院理学研究科)
タ イ ト ル: Cross-exchangeable cycles and 1-handles for surface diagrams

A surface-knot is an embedded closed oriented surface in 4-space.
A surface diagram is the image of a projection of a surface-knot into 3-space with crossing information.
For every surface-knot, we can attach some 1-handles to obtain a trivial surface. The minimal number of such 1-handles is called the unknotting number of the surface-knot.
Some surface diagrams have special double curves, in which we can change the crossing information to obtain a trivial surface.
In this talk we will discuss about a relation between the number of such special double curves, which may have multiple points and the number of 1-handles needed to obtain a trivial surface.

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講 演 者: Tom Fleming(University of California, San Diego (UCSD))
タ イ ト ル: Chirality of Alternating Knots in S x I

Many alternating knots, such as the figure-eight knot, are isotopic to their mirror image in $S^{3}$. Detecting amphichiral knots in $S^{3}$ can be difficult.  The situation for alternating knots embedded in a surface cross an interval is quite different. There is a well defined projection in a such a space, and we define the mirror image of a knot to be the reflection across the projection surface.  We can then show if an alternating knot K is non-trivially embedded in S x I, where the genus of S is greater than zero, then K is not isotpic to its mirror image. In other words, such a knot is chiral.

We prove this result by studying the span of a generalized Kauffman polynomial of the knot, as well as the polynomial of lifts of the knot in certain covering spaces of S.  The condition of "nontrivially embedded" is necessary to avoid the obvious counterexample induced by embedding a small $S^{3}$ into S x I.

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講 演 者: 田中 利史(東京大学大学院数理科学研究科,COE研究員)
タ イ ト ル: Ribbon 2-knots assoiciated with symmetric unions

A symmetric union was introduced by Kinoshita and Terasaka in the 1950's, which is a generalization of the connected sum operation for a knot and its mirror image. It has been generalized by Lamm recently. Every symmetric union is a ribbon knot and it has been shown that a ribbon knot with crossing number less than or equal to ten is a symmetric union. We ask if every ribbon 2-knot has a symmetric union as an equatorial cross section (i.e. every ribbon 2-knot is assoiciated with symmetric union) because every 2-knot has a ribbon knot as an equatorial cross section. In this talk, we prove that every ribbon 2-knot of 1-fusion is assoiciated with a symmetric union for the unknot or a 2-bridge knot. We also introduce a banded symmetric union to study a ribbon 2-knot and we generalize a result of Kanenobu concerning some family of ribbon 2-knots.

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講 演 者: 田山 育男(大阪市立大学数学研究所)
タ イ ト ル: Enumerating the exteriors of prime links by a canonical order

This is a joint work with Akio Kawauchi. He gave a well-order to the set of links, which induces the well-orders into the set of link exteriors and the set of 3-manifolds. The length of a link is the minimum string number when we deform the link into a closed braid. The length of a link exterior and that of a 3-manifold is defined by using the length of a link. We enumerated the prime links with up to length 10. Our goal of this study is to enumerate    3-manifolds with up to length 10. In this talk, we enumerate the exteriors of prime links with up to length 9, which is in a step of the goal.

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講 演 者: 鈴木 正明(東京大学数理)
タ イ ト ル: Twisted Alexander invariant and a partial order in the knot table

Twisted Alexander invariant is defined for a finitely presentable group $G$ and a representation of $G$ and a surjective homomorphism of $G$ to a free abelian group. In this talk, we introduce some examples and some properties of the twisted Alexander invariant. Moreover, as an application, we consider a partial order on the set of prime knots. 
Let $K$ be a knot and $G(K)$ the knot group. For two prime knots $K,K'$, we write $K \geq K'$, if there exists a surjective homomorphism from $G(K)$ to $G(K')$. We determine this partial order ``$\geq$'' on the set of prime knots in the Rolfsen's table.

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講 演 者: 市原 一裕(大阪産業大学教養部)
タ イ ト ル: Alexander polynomials of doubly primitive knots
    (joint work with Toshio Saito (Osaka University)
     and Masakazu Teragaito (Hiroshima University))

It is an interesting open problem to determine the knots in the 3-sphere admitting a lens space surgery. Known such examples are only knots called `doubly primitive', and it is conjectured that they are all. Recently, the Alexander polynomials of such knots have been studied by Ozsvath and Szabo, Kadokami, and Yamada. In this talk, we give a formula for Alexander polynomials of doubly primitive knots. This also gives a practical algorithm to determine the genus of any doubly primitive knot.

Toptop



講 演 者: 佐藤 進(千葉大学大学院自然科学研究科)
タ イ ト ル: Disk presentations of surface-knots and -links

We introduce a new way of presenting surface-knots and -links in $S^4$, called disk presentation. This enables us to define the disk index $¥Delta(F)$ of a surface-knot or -link $F$. We prove that $¥Delta(F)=2$ if and only if $F$ is a trivial $S^2$-knot, and $¥Delta(F)=3$ if and only if $F$ is a trivial non-orientable surface-knot with $|e(F)| ¥leq  3-¥chi(F)$, where $e(F)$ is the normal Euler number of $F$ and $¥chi(F)$ is the Euler characteristic of $F$.

Toptop


講 演 者: 中西 清孝(九州大学大学院数理学府)
タ イ ト ル: Sliding relation in the Kauffman bracket skein module of a 2-bridge knot exterior

In this talk, we give an expicit formula for an underlying relation in the Kauffman bracket skein module of the exterior of the $2$-bridge knot $S(2pq+1, 2q)$, which is called the ``sliding relation" in this talk. This relation comes from a sliding of the trivial knot in a genus $2$ handlebody along a simple closed curve (attaching slope) along which a $2$-handle is attached to obtain the exterior of a $2$-bridge knot, and we show that the relation is essential. We also give an application of the formula to $SL(2, \mathbb{C})$ character variety of the fundamental group of the $2$-bridge knot complement. Finally we research the commensurability of such $2$-bridge knot exteriors.

Toptop


講 演 者: 石井 敦(大阪大学大学院理学研究科)
タ イ ト ル: Skein relations for the generalized Alexander polynomial for virtual
links and closed virtual 2-braids

We show how to find a skein relation for the generalized Alexander polynomial for virtual links.
   
A quantum invariant for classical links is defined by associating a vector space to a strand. A linear relation among linear maps associated to oriented classical tangles is a skein relation which is helpful for evaluating the invariant. Since a linear map associated to an oriented classical tangle is an intertwiner which is equivariant with respect to the action on $V \otimes V$, we may find a skein relation among any $n$ oriented classical tangles if $n$ is greater than the dimension of the space of intertwiners.
   
On the other hand, a quantum invariant for virtual links is in the different situation from above. A linear map associated to an oriented virtual tangle is not an intertwiner any more. So we might appreciate an alternative way to find a skein relation for a quantum invariant for virtual links. We focus on the generalized Alexander polynomial for virtual links. We introduce a finite dimensional vector space which includes all linear maps associated to oriented virtual tangles, and gave the dimension of the vector space.
   
Furthermore, by using this vector space, we find a new skein relation. As an application of the relation, we give a formula for the invariant for a closed virtual $2$-braid.

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最終更新日: 2006年1月12日
(C)大阪市大数学教室