大阪市立大学数学研究所 (Osaka City University Advanced Mathematical Institute)

 Friday Seminar on Knot Theory（2013年度） 2012年度
2013年度組織委員　 岡崎 真也

キャンパスマップ

 講 演 者 ：Gyo Taek Jin (KAIST) タ イ ト ル ：Quadrisecants of unknots （アブストラクト） (PDF) 日 時 ：2月21日（金）16：30～17：15 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：滝岡 英雄 (大阪市立大学) タ イ ト ル ：The $\Gamma$-polynomial of a knot and its applications （アブストラクト） (PDF) 日 時 ：2月21日（金）15：30～16：15 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：Hwa Jeong Lee (KAIST) タ イ ト ル ：On the arc index of knots and links （アブストラクト） (PDF) 日 時 ：2月21日（金）14：30～15：15 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：岡崎 真也 (大阪市立大学数学研究所) タ イ ト ル ：Seifert manifolds and $0$-surgery （アブストラクト） (PDF) 日 時 ：2月21日（金）13：30～14：15 場 所 数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：金信 泰造 (大阪市立大学) タ イ ト ル ：H(2)-Move and Other Local Moves on Knots （アブストラクト） (PDF) 日 時 ：1月17日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：井上 歩 (愛知教育大学) タ イ ト ル ：Colorings of torus knots and PL trochoids （アブストラクト） (PDF) 日 時 ：1月10日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：張 娟姫 (奈良女子大学) タ イ ト ル ：Bridge splittings of links with Hempel distance $n$ （アブストラクト） (PDF) 日 時 ：12月13日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：大城 佳奈子 (上智大学) タ イ ト ル ：Linear Alexander quandle colorings and finite-fold cyclic covers of $S^3$ branched over knots （アブストラクト） (PDF) 日 時 ：11月29日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：Philippe Humbert (University of Strasbourg) タ イ ト ル ：Higher genus tangles （アブストラクト） (PDF) 日 時 ：11月29日（金）15：00～16：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：安原 晃 (東京学芸大学) タ イ ト ル ：$C_k$-concordance group of $n$-string links (joint work with Jean-Baptiste Meilhan (University of Grenoble I)) （アブストラクト） (PDF) 日 時 ：11月22日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：田中 心 (東京学芸大学) タ イ ト ル ：Regular-equivalence of 2-knot diagrams and sphere eversions （アブストラクト） (PDF) 日 時 ：11月22日（金）15：00～16：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：平澤 美可三 (名古屋工業大学) タ イ ト ル ：A generalization of the Murasugi sum of Seifert surfaces （アブストラクト） (PDF) 日 時 ：11月8日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：小鳥居 祐香 (東京工業大学) タ イ ト ル ：The relation between Milnor mu-invariant and HOMFLYPT polynomial for links （アブストラクト） (PDF) 日 時 ：11月1日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：矢口 義朗 (群馬工業高等専門学校) タ イ ト ル ：Cords on a 3-times punctured disk （アブストラクト） (PDF) 日 時 ：10月18日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：安部 哲哉 (東京工業大学・学振PD) タ イ ト ル ：Infinitely many ribbon disks with the same exterior （アブストラクト） (PDF) 日 時 ：10月11日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：伊藤 昇 (早稲田大学高等研究所) タ イ ト ル ：(1, 2), weak (1, 3), and strong (1, 3) homotopies on knot projections （アブストラクト） (PDF) 日 時 ：10月4日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：中村 伊南沙 (学習院大学・学振PD) タ イ ト ル ：Triple point cancelling numbers of torus-covering knots （アブストラクト） (PDF) 日 時 ：7月19日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：高尾 和人 (広島大学) タ イ ト ル ：Destabilized bridge spheres of knots （アブストラクト） (PDF) 日 時 ：7月12日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：梅本 悠莉子 (大阪市立大学) タ イ ト ル ：Growth rates of cocompact hyperbolic Coxeter groups and 2-Salem numbers （アブストラクト） (PDF) 日 時 ：7月5日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：屋代 司 (Sultan Qaboos Univeristy) タ イ ト ル ：Constructing surface-diagrams with cross-exchangeable cycles （アブストラクト） (PDF) 日 時 ：6月28日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：伊藤 哲也 (京都大学　数理解析研究所) タ イ ト ル ：Singular spanning discs, framing function of knots, and strength version of Dehn's lemma （アブストラクト） (PDF) 日 時 ：6月21日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：佐藤 進 (神戸大学) タ イ ト ル ：On knots with no 3-state （アブストラクト） (PDF) 日 時 ：6月14日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：櫻井 みぎ和 (東京女子大学) タ イ ト ル ：An estimate of the unknotting numbers for virtual knots by forbidden moves （アブストラクト） (PDF) 日 時 ：6月7日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：森内 博正 (大阪市立大学 数学研究所 (OCAMI)) タ イ ト ル ：A table of coherent band-Gordian distances between knots （アブストラクト） (PDF) 日 時 ：5月31日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：早野 健太 (大阪大学) タ イ ト ル ：On four-manifolds with genus-1 simplified broken Lefschetz fibrations （アブストラクト） (PDF) 日 時 ：5月10日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：鎌田 聖一 (大阪市立大学) タ イ ト ル ：Chart descriptions of 2-dimensional braids （アブストラクト） (PDF) 日 時 ：4月26日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：秋吉 宏尚 (大阪市立大学) タ イ ト ル ：Hyperbolic structures on the torus with a single cone point （アブストラクト） (PDF) 日 時 ：4月19日（金）16：00～17：00 場 所 ：数学　第3セミナー室（共通研究棟4階401室） Top 講 演 者 ：滝岡 英雄 (大阪市立大学) タ イ ト ル ：The cable $\Gamma$-polynomial of a knot （アブストラクト） (PDF) 日 時 ：4月12日（金）16：00～17：00 場 所 ：数学　第３セミナー室（共通研究棟4階401室） Top

 アブストラクト集

 講 演 者： Gyo Taek Jin (KAIST) タ イ ト ル： Quadrisecants of unknots

The conjecture of quadrisecant approximation is ture for some minimal polygonal prime knots. We investigate the conjecture for polygonal unknots. We thank Ernst Claus for his data of polygonal unknots.

 講 演 者： 滝岡 英雄 (大阪市立大学) タ イ ト ル： The $\Gamma$-polynomial of a knot and its applications

The $\Gamma$-polynomial is an invariant of an oriented link in the 3-sphere, which is contained in both the HOMFLYPT and Kauffman polynomials as their common zeroth coefficient polynomial. As applications of the $\Gamma$-polynomial, I will talk about the following three topics:
(1) On the arc index of cable knots (joint with Hwa Jeong Lee, KAIST)
(2) On the braid index of Kanenobu knots
(3) On the arc index of Kanenobu knots (joint with Hwa Jeong Lee, KAIST)

 講 演 者： Hwa Jeong Lee (KAIST) タ イ ト ル： On the arc index of knots and links

Every knot or link $L$ can be embedded in the union of finitely many half planes which have a common boundary line such that each half plane intersects $L$ in a single arc. Such an embedding is called an arc presentation of $L$. The arc index of $L$ is the minimal number of pages among all arc presentations of $L$. It is known that the arc index of a knot is closely related to the minimal crossing number of the knot. In this talk, we present a small survey on arc index and compute the arc index of some of Pretzel knots and Montesinos links.

 講 演 者： 岡崎 真也 (大阪市立大学数学研究所) タ イ ト ル： Seifert manifolds and $0$-surgery

For closed connected orientable $3$-manifold $M$, let $c(M)$ be the minimal number of the component number of any link $L$ whose each component is the unknot in $S^3$ such that $M$ is obtained by the $0$-surgery of $S^3$ along $L$. Then $c(M)$ is an invariant of closed connected orientable $3$-manifold $M$. We have already obtained $c(M)$ for some lens spaces. In this talk, we consider some Seifert manifolds obtained by the $0$-surgery of $S^3$ along a pure $3$-braid link, and we determine $c(M)$ for some Seifert manifolds. Moreover, we calculate the bridge genus and the braid genus for some Seifert manifolds.

 講 演 者： 金信 泰造 (大阪市立大学) タ イ ト ル： H(2)-Move and Other Local Moves on Knots

An H(2)-move is a local move on an unoriented knot which is realized by smoothing a crossing. This is an unknotting operation, that is, any knot can be unknotted by a sequence of H(2)-moves. So, we may define an H(2)-unknotting number and H(2)-Gordian distance. We introduce several methods to give a lower bound of the H(2)-Gordian distance, which allow us to improve the table of H(2)-Gordian distances for knots with up to seven crossings. We also consider a relation with the band surgery and delta move.

 講 演 者： 井上 歩 (愛知教育大学) タ イ ト ル： Colorings of torus knots and PL trochoids

The set consisting of rotations of the Euclidean plane is obviously equipped with the structure of a quandle. In this talk, we show that we have a non-trivial coloring of a torus knot by the quandle related to a PL trochoid.

 講 演 者： 張 娟姫 (奈良女子大学) タ イ ト ル： Bridge splittings of links with Hempel distance $n$

Hempel distance of bridge splittings of links is a measurement of certain complexity of bridge splittings. The distance is known to reflect some topological and geometric properties of bridge splittings and links themselves. In this talk, we show the existence of bridge splittings of links with Hempel distance exactly $n$ for any given integer $n$. This is a joint work with Ayako Ido and Tsuyoshi Kobayashi.

 講 演 者： 大城 佳奈子 (上智大学) タ イ ト ル： Linear Alexander quandle colorings and finite-fold cyclic covers of $S^3$ branched over knots

The Fox-colorings of a knot are interpreted as the group representations from the fundamental groups of the $2$-fold cyclic cover of $S^3$ branched over the knot to $\mathbb Z_p$. The interpretation is extended for linear Alxander quandle colorings by using some condition.

 講 演 者： Philippe Humbert (University of Strasbourg) タ イ ト ル： Higher genus tangles

This talk will be about tangles (knots, braids...) lying in a cylinder over a closed surface of arbitrary genus. I will first introduce some kind of planar diagrams and Reidemeister-like moves for these objects. This diagrammatic point of view will then lead to a universal property stated in the language of braided categories.

 講 演 者： 安原 晃 (東京学芸大学) タ イ ト ル： $C_k$-concordance group of $n$-string links (joint work with Jean-Baptiste Meilhan (University of Grenoble I))

The $C_k$-equivalence is an equivalence relation on $n$-string links which is genarated by $C_k$-move and concordance. The set of $C_k$-concordance classes of $n$-string links has a group structure. We decide when the quotient groups become abelian. In particular, we show that the $C_9$-concordance group of 2-string links is not abelian.

 講 演 者： 田中 心 (東京学芸大学) タ イ ト ル： Regular-equivalence of 2-knot diagrams and sphere eversions

A surface-knot diagram is said to be regular if it has no branch points. In this talk, we construct two regular diagrams of a $2$-knot such that any sequence of Roseman moves between them involves branch points. This is a joint work with Masamichi Takase (Seikei University).

 講 演 者： 平澤 美可三 (名古屋工業大学) タ イ ト ル： A generalization of the Murasugi sum of Seifert surfaces

The Murasugi sum is a natural operation to glue two Seifert surfaces. Let F = G ＊ H be a Murasugi sum of two Seifert surfaces G and H. Then the following are well-known:
(i) F is of minimal genus if and only if so are G and H.
(ii) F is a fiber surface if and only if so are G and H.

In this talk, we generalize the notion of Murasugi sum by using surfaces other than a disk, and show that the operation also enjoys the above-mentioned properties.

Neumann & Rudolph have introduced the notion of "unfoldings" in n-dimensional knot theory. However, in case n=3, all known examples of unfoldings are realized as decompositions of Murasugi sums. We give examples of our operation which are not Murasugi sums or "unfoldings". After formulating the gap between out operation and the Murasugi sum, we show that the gap can be arbitrarily large.

 講 演 者： 小鳥居 祐香 (東京工業大学) タ イ ト ル： The relation between Milnor mu-invariant and HOMFLYPT polynomial for links

This is joint work with Akira Yasuhara (Tokyo Gakugei University).
For an ordered, oriented link in the 3-sphere, J. Milnor defined a family of invariants, known as Milnor $\overline{\mu}$-invariants. For an $n$-component link, Milnor invariant is specified by a sequence of elements of $\{1, 2, \ldots, n \}$ and the length of the sequence is called the length of the Milnor invariant. J.-B. Meilhan and A. Yasuhara showed that any Milnor $\overline{\mu}$-invariant of length between 3 and $2k+1$ can be represented as a combination of HOMFLYPT polynomial of knots obtained by certain band sum of the link components, if all $\overline{\mu}$-invariants of length $\leq k$ vanish. In this talk, we improve their formula to give the $\overline{\mu}$-invariants of length $2k+2$ by adding correction terms. The correction terms can be given by a combination of HOMFLYPT polynomial of knots determined by $\overline{\mu}$-invariants of length $k+1$. In particular, for any 4-component link the $\overline{\mu}$-invariants of length 4 are given by our formula, since all $\overline{\mu}$-invariants of length 1 vanish.

 講 演 者： 矢口 義朗 (群馬工業高等専門学校) タ イ ト ル： Cords on a 3-times punctured disk

A cord is a simple curve on a punctured disk, which connects two punctures. In this talk, we introduce diagrams which represent isotopy classes of cords. Using such diagrams, we make up a list of all isotopy classes of cords on a 3-times punctured disk. As a result, it is shown that they are completely parameterized by 3 non-negative integers.

 講 演 者： 安部 哲哉 (東京工業大学・学振PD) タ イ ト ル： Infinitely many ribbon disks with the same exterior

In 1962, Gluck proved that there are, at most, two different 2-knots with the same exterior. In 1976, Gordon proved that there exist two different 2-knots with the same exterior.

In this talk, we consider an analogues problem for ribbon disks in the 4-ball D^4. We observe that there exist infinitely many ribbon disks with the same exterior. This result follows from the previous joint work with M.Tange. We also study whether the exterior is a handlebody bundle over S^1.

 講 演 者： 伊藤 昇 (早稲田大学高等研究所) タ イ ト ル： (1, 2), weak (1, 3), and strong (1, 3) homotopies on knot projections

The speaker plans to talk about a joint work with Yusuke Takimura (Waseda University, School of Education, M2). First, we obtain the necessary and sufficient condition that when two knot projections are related by a finite sequence of the first and second flat Reidemeister moves. Second, we introduce weak (1, 3) homotopy that is an equivalence relation on knot projections, defined by the first flat Reidemeister move and one of the third flat Reidemeister moves. Third, using a map sending weak (1, 3) homotopy classes to knot isotopy classes, we determine which knot projections are trivialized under weak (1, 3) homotopy.

If time permits, the speaker will discuss another joint work with Y. Takimura and K. Taniyama. The joint work introduces strong (1, 3) homotopy that is an equivalence relation on knot projection, defined by the first flat Reidemeister move and another type of the third flat Reidemeister moves. Showing that Hanaki's trivializing number is weak (1, 3) invariant and introducing cross chord numbers that produce a strong (1, 3) invariant, we claim that two knot projections having trivializing number two are weak (1, 3) homotopy equivalent and strong (1, 3) homotopy equivalent if and only if the two knot projections with trivializing number two can be related by only the first flat Reidemeister moves. We also determine the strong (1, 3) equivalence class containing the trivial knot projection and other classes of knot projections.

 講 演 者： 中村 伊南沙 (学習院大学・学振PD) タ イ ト ル： Triple point cancelling numbers of torus-covering knots

It is known that any surface knot can be transformed to an unknotted surface knot or a surface knot which has a diagram with no triple points by a finite number of 1-handle additions. The minimum number of such 1-handles is called the unknotting number or the triple point cancelling number, respectively. In December 2011, I gave a talk in this seminar on upper bounds and lower bounds of unknotting numbers of torus-covering knots, which are surface knots in the form of coverings over the standard torus $T$. In this talk, we give lower bounds of triple point cancelling numbers of torus-covering knots, by using Iwakiri's result and calculating quandle cocycle invariants. In particular, we give examples of torus-covering knots whose unknotting numbers and triple point cancelling numbers are exactly two.

 講 演 者： 高尾 和人 (広島大学) タ イ ト ル： Destabilized bridge spheres of knots

Any knot admits infinitely many bridge spheres, and to classify them is a general problem. Destabilized bridge spheres are of particular interest because all the other can be obtained from them by stabilizations up to isotopy. In this talk, we introduce a criterion which guarantees a bridge sphere to be destabilized, and give a knot which has destabilized bridge spheres of bridge number arbitrarily higher than the bridge number of the knot. This is a joint work with Yeonhee Jang, Tsuyoshi Kobayashi and Makoto Ozawa.

 講 演 者： 梅本 悠莉子 (大阪市立大学) タ イ ト ル： Growth rates of cocompact hyperbolic Coxeter groups and 2-Salem numbers

The group generated by reflections with respect to facets of a Coxeter polytope in n-dimensional hyperbolic space $\\H^n$ is called a hyperboric Coxeter group. By the results of Cannon, Wagreich and Parry, it is known that the growth rate of a cocompact Coxeter group in $\H^2$ and $\H^3$ is a Salem number. On the other hand, Kerada defined a $j$-Salem number, which is a generalization of a Salem number. In this talk, I will present that we realize infinitely many 2-Salem numbers as the growth rates of cocompact Coxeter groups in $\H^4$. Our Coxeter polytopes are constructed by successive gluing of Coxeter polytopes which we call Coxeter dominoes.

 講 演 者： 屋代 司 (Sultan Qaboos Univeristy) タ イ ト ル： Constructing surface-diagrams with cross-exchangeable cycles

Roseman moves are local deformations of surface diagrams which are generalized version of Reidemeister moves of knot diagrams. Each Roseman move requires geometric conditions. We look at the the move which involves a saddle and a regular disc. This move changes the number of immersed circles or immersed intervals in the double decker set. For some diagrams we cannot apply this move to obtain a different diagram. We call this diagram a d-minimal surface diagram. On the other hand, we can define a special double curve in a surface diagram along which we can change the　crossing information so that we obtain a trivial diagram. We call this curve a cross-exchangeable cycle or arc. In this talk we present a construction of a series of d-minimal surface diagrams with cross-exchangeable cycles.
This research is a joint work with Abdul Mohamad.

 講 演 者： 伊藤 哲也 (京都大学　数理解析研究所) タ イ ト ル： Singular spanning discs, framing function of knots, and strength version of Dehn's lemma

Greene-Wiest introduced a framing function of K by counting the intersection of a singular disc spanned by K. In this talk we explain basics of framing functions emphasizing interactions with several aspects of knot theory. We show a lower bound of framing functions and as an application, we give a slightly generalized version of Dehn's lemma.

 講 演 者： 佐藤 進 (神戸大学) タ イ ト ル： On knots with no 3-state

Kauffman introduces the state model for a Jones polynomial, where the number of circles in each state is an important quantity. For a positive integer k, a k-state of a (classical or virtual) knot diagram is such a state consisting of k circles. It is easy to see that any non-trivial diagram has 1- and 2-states both. In this talk, we determine knot diagrams with no 3-states via Gauss diagrams, and give several properties related to the integer-writhes, upper and lower knot groups, and Miyazawa polynomials.

 講 演 者： 櫻井 みぎ和 (東京女子大学) タ イ ト ル： An estimate of the unknotting numbers for virtual knots by forbidden moves

It is known that any virtual knot can be deformed into the trivial knot by a finite sequence of forbidden moves. In this talk, we give the difference of the values obtained from some invariants constructed by A. Henrich between two virtual knots which can be transformed into each other by a single forbidden move. As a result, we obtain a lower bound of the unknotting number of a virtual knot by forbidden moves.

 講 演 者： 森内 博正 (大阪市立大学 数学研究所 (OCAMI)) タ イ ト ル： A table of coherent band-Gordian distances between knots

A coherent band surgery is a local move on an oriented link, which is equivalent to a smoothing a crossing. The coherent band-Gordian distance between two links is the least number of coherent band surgeries needed to transform one link into the other. We introduce some criteria for two links which are related by a coherent band surgery. Then we give a table of coherent band-Gordian distances between two knots with up to seven crossings.
This is a joint work with Taizo Kanenobu.

 講 演 者： 早野 健太 (大阪大学) タ イ ト ル： On four-manifolds with genus-1 simplified broken Lefschetz fibrations

In 2005, Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations in order to understand near-symplectic structures via fibration structures. Simplified broken Lefschetz fibrations are broken Lefschetz fibrations with several conditions on topology and configuration of singularities. Although negative definite four-manifolds cannot admit near-symplectic structures, it turns out that every closed, oriented, connected four-manifold has a simplified broken Lefschetz fibration. In this talk, we first relate simplified broken Lefschetz fibrations to mapping class groups via monodromy representations. Using this relation, we then discuss the classification problem of genus-1 simplified broken Lefschetz fibrations.

 講 演 者： 鎌田 聖一 (大阪市立大学) タ イ ト ル： Chart descriptions of 2-dimensional braids

The chart description was first introduced by the speaker to describe simple 2-dimensional braids. In this talk we consider chart descriptions for non-simple 2-dimensional braids, especially those called "regular". Any regular 2-dimensional braid can be described by a regular chart, and such regular descriptions are related by certain moves.

 講 演 者： 秋吉 宏尚 (大阪市立大学) タ イ ト ル： Hyperbolic structures on the torus with a single cone point

We construct hyperbolic structures on the torus with a single cone point in a canonical way. It is proved that a variant of McShane's identity holds for such a structure by Tan-Wong-Zhang, where they developed the study on generalized Markoff maps and showed that the Bowditch's Q-Condition (BQ-condition) is crucial for the convergence of the identity. Our proof uses their results to find a canonical generators for a given real generalized Markoff map satisfying the BQ-condition.

 講 演 者： 滝岡 英雄 (大阪市立大学) タ イ ト ル： The cable $\Gamma$-polynomial of a knot

The $\Gamma$-polynomial is an invariant of an oriented link, which is the zeroth coefficient polynomial of both the HOMFLYPT polynomial and the Kauffman polynomial. In particular, we study the cable $\Gamma$-polynomial of a knot, that is, the $\Gamma$-polynomial of a cable knot. I will talk about several results of the 2-cable $\Gamma$-polynomials of the Kanenobu knots and the 3-cable $\Gamma$-polynomial of a mutant knot.

 最終更新日: 2014年2月12日 (C)大阪市大数学教室