市大数学教室

大阪市立大学数学研究所
(Osaka City University Advanced Mathematical Institute)
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Friday Seminar on Knot Theory(2014年度)
2013年度
2014年度組織委員  岡崎 真也

講 演 者 :村上 斉 (東北大学)
タ イ ト ル :The volume conjecture and its generalizations
(アブストラクト) (PDF)
日 時 :12月12日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :Alexander Zupan (University of Texas at Austin)
タ イ ト ル :Surface knots with small bridge number
(アブストラクト) (PDF)
日 時 :12月5日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :金城 就実 (信州大学)
タ イ ト ル :Immersions of 3-sphere into 4-space associated with Dynkin
diagrams of types A and D
(アブストラクト) (PDF)
日 時 :11月28日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :福永 知則 (九州産業大学)
タ イ ト ル :Truncated Polyak algebras of Gauss words and
classification of Gauss words by finite type invariants.
(アブストラクト) (PDF)
日 時 :11月7日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :小林 竜馬 (東京理科大学)
タ イ ト ル :A finite presentation of the level 2 principal congruence
subgroup of GL(n;Z) and its application
(アブストラクト) (PDF)
日 時 :10月10日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :林 晋 (東京大学)
タ イ ト ル :Localization of Dirac Operators on 4n+2 Dimensional Open
Spin^c Manifolds and Its Applications.
(アブストラクト) (PDF)
日 時 :10月3日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :Arnaud Mortier (OCAMI)
タ イ ト ル :Cohomology of arrow diagram spaces
(アブストラクト) (PDF)
日 時 :7月18日(金)16:40~17:40
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :Victoria Lebed (OCAMI)
タ イ ト ル :Towards braid-theoretic applications of Laver tables
(アブストラクト) (PDF)
日 時 :7月18日(金)15:30~16:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :鄭 仁大 (近畿大学)
タ イ ト ル :Annulus twist and diffeomorphic 4-manifolds II
(アブストラクト) (PDF)
日 時 :7月11日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :佐藤 光樹 (東京学芸大学)
タ イ ト ル :Non-orientable genus of a knot in punctured Spin 4-manifolds
(アブストラクト) (PDF)
日 時 :7月4日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :岩切 雅英 (佐賀大学)
タ イ ト ル :Unknoting numbers for handlebody-knots and Alexander
quandle colorings
(アブストラクト) (PDF)
日 時 :6月27日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :屋代 司 (Sultan Qaboos University)
タ イ ト ル :Lower decker sets and triple points for surface-knots
(アブストラクト) (PDF)
日 時 :6月20日(金)16:40~17:40
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :Amal Al Kharusi (Sultan Qaboos University)
タ イ ト ル :Unknotting operation and independent components of lower
decker set
(アブストラクト) (PDF)
日 時 :6月20日(金)15:30~16:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :Daniele Zuddas (Korea Institute for Advanced Study)
タ イ ト ル :An equivalence theorem for Lefschetz fibrations over the disk
(アブストラクト) (PDF)
日 時 :6月13日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :藤井 道彦 (京都大学)
タ イ ト ル :On minimal crossing expressions of braids
(アブストラクト) (PDF)
日 時 :6月6日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :Fedor Duzhin (Nanyang Technological University)
タ イ ト ル :Brunnian braids and Brunnian links
(アブストラクト) (PDF)
日 時 :5月30日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :鎌田 直子 (名古屋市立大学)
タ イ ト ル :Writhes of a twisted knot derived from weighted index diagrams
(アブストラクト) (PDF)
日 時 :5月16日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :浮田 卓也 (東京工業大学)
タ イ ト ル :Genus zero PALF structures on the Akbulut cork and exotic
pairs
(アブストラクト) (PDF)
日 時 :5月9日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :田中 利史 (岐阜大学)
タ イ ト ル :On the maximal Thurston-Bennequin number of knots and
links in a spatial graph
(アブストラクト) (PDF)
日 時 :5月2日(金)16:30~17:30
場 所 :数学 第3セミナー室(共通研究棟4階401室)
Toptop
講 演 者 :門田 直之 (大阪電気通信大学)
タ イ ト ル :Non-holomorphic Lefschetz fibrations with (-1)-sections
(アブストラクト) (PDF)
日 時 :4月25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)
Toptop
講 演 者 :河村 建吾 (大阪市立大学)
タ イ ト ル :On the clasp number of a knot
(アブストラクト) (PDF)
日 時 :4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)
Toptop
講 演 者 :植木 潤 (九州大学)
タ イ ト ル :On the universal deformations of SL(2)-representations of
2-bridge knot groups
(アブストラクト) (PDF)
日 時 :4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)
Toptop



アブストラクト集



講 演 者: 村上 斉 (東北大学)
タ イ ト ル: The volume conjecture and its generalizations

I will talk about a relation of the colored Jones polynomial of a knot to the Chern-Simons invariant, which is a kind of complex volume, and the Reidemeister torsion associated with a representation of the knot group to SL(2;C).

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講 演 者: Alexander Zupan (University of Texas at Austin)
タ イ ト ル: Surface knots with small bridge number

Recently, Gay and Kirby introduced a 4-dimensional analogue of a Heegaard splitting called a trisection. We adapt their approach to show that every knotted surface in the 4-sphere admits a bridge trisection; namely, there is a decomposition of the 4-sphere into three 4-balls which splits the surface into a collection of boundary parallel disks. This may be viewed as the 4-dimensional version of a bridge decomposition for a knot in the 3-sphere, and as such, a bridge trisection has two complexity parameters akin to the bridge number of a knot. We will discuss a classification of bridge trisections with relatively low complexity. This talk is based on joint work in progress with David Gay and Jeffrey Meier.

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講 演 者: 金城 就実 (信州大学)
タ イ ト ル: Immersions of 3-sphere into 4-space associated with Dynkin
diagrams of types A and D

The Smale--Hirsch h-principle implies that the group of regular homotopy classes of immersions of m-sphere into N-space is isomorphic to the m th homotopy group of the Stiefel manifold of m frames in N-space. The isomorphism is given as taking the differential at each point of m-sphere and is called the Smale invariant. In particular, Ekholm and Takase have given a formula for the Smale invariant of an immersion of 3-sphere into 4-space by using a singular Seifert surface for the immersion.

In this talk we will explain the construction of two infinite sequences of immersions of the 3-sphere into 4-space, parametrized by the Dynkin diagrams of types A and D. The construction is based on immersions of 4-manifolds obtained as the plumbed immersions along the weighted Dynkin diagrams. We will compute their Smale invariants by using Ekholm--Takase's formula in terms of singular Seifert surfaces.

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講 演 者: 福永 知則 (九州産業大学)
タ イ ト ル: Truncated Polyak algebras of Gauss words and
classification of Gauss words by finite type invariants.

In this talk, we study the universal finite type invariants of Gauss words which was introduced by N. Ito and A. Gibson in [1]. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degree 5, 6 and 7 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank less than or equal to 5. This talk is based on a joint work with T. Yamaguchi (Hiroshima University) and T. Yamanoi (Hokkaido university).

[1] N. Ito and A. Gibson, "Finite type invariants of nanowords and nanophrases" Topol. Appl. 158 (2011), 1050 -- 1072.

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講 演 者: 小林 竜馬 (東京理科大学)
タ イ ト ル: A finite presentation of the level 2 principal congruence
subgroup of GL(n;Z) and its application

Let $\Gamma_2(n)$ be the kernel of the homomorphism from $GL(n;Z)$ to $GL(n;Z/2Z)$. Note that for an element $A$ in $\Gamma_2(n)$, the diagonal entries of $A$ are odd, the other entries are even. A finite generating set of $\Gamma_2(n)$ has been known. In our work, we obtained a finite presentation of $\Gamma_2(n)$. To obtain a presentation, we constructed a simply connected simplicial complex which $\Gamma_2(n)$ acts on.
In this talk, we will introduce this complex. We note that a presentation of $\Gamma_2(n)$ has been independently obtained also by Fullarton and Margalit-Putman recently. As an application, we obtained a generating set of the Torelli group of a non-orientable closed surface. We will also talk about it.

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講 演 者: 林 晋 (東京大学)
タ イ ト ル: Localization of Dirac Operators on 4n+2 Dimensional Open
Spin^c Manifolds and Its Applications.

We observe that, on a 4n+2 dimensional Spin^c manifold which is not necessarily closed, the symbol of the Dirac operator can be localized by a section of its determinant line bundle to the neighborhood of its zero set. If the zero set is compact, an integer-valued topological index of the Dirac operator can be defined, which gives the generalization of the usual index defined on a closed 4n+2 dimensional Spin^c manifold.

As an application of this localization method, two well known results will be shown. One is a relation between the index and an index of a Dirac operator of its characteristic submanifold. This formula was proved by J. Fast and S. Ochanine for even dimensional closed Spin^c manifolds by a localization of K-class. We prove this formula for 4n+2 dimensional Spin^c manifolds in a different way. The other is the Riemann-Roch theorem for the Riemann surface with boundary. We prove this formula in a topological way.

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講 演 者: Arnaud Mortier (OCAMI)
タ イ ト ル: Cohomology of arrow diagram spaces

Vassiliev invariants of knots are the 0-codimensional part of a theory of "finite-type" cohomology which, apart from the level 0, is mostly unknown. Among the computational methods at the level 0, the most efficient may be M.Polyak and O.Viro's formulas, represented by linear combinations of arrow diagrams. Such a combination defines an invariant if and only if it lies in the kernel of a certain linear map, and it is conjectured that this linear map can be enhanced into a full cochain complex, whose i-th homology describes Vassiliev i-cocycles. In this talk, I will show that the first step of this conjecture is true, give examples of 1-cocycles in the space of knots, and show how to evaluate them on the two main cycles in the space of knots, namely Gramain's and Hatcher's loops.

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講 演 者: Victoria Lebed (OCAMI)
タ イ ト ル: Towards braid-theoretic applications of Laver tables

Laver tables (LT) are certain finite shelves (i.e., sets endowed with a binary operation distributive with respect to itself). They originate from Set Theory. In spite of an elementary definition, LT have complicated combinatorial properties. Conjecturally, they are finite approximations of the free monogenerated shelf, whose spectacular applications to Braid Theory are well studied. It is thus natural to expect interesting topological applications of LT as well. This talk is devoted to our first steps in this direction. Namely, we describe all 2- and 3-cocycles for LT, study their promisingly rich structure, and discuss applications to Braid Theory.

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講 演 者: 鄭 仁大 (近畿大学)
タ イ ト ル: Annulus twist and diffeomorphic 4-manifolds II

We show that for any integer $n$, there exist infinitely many mutually distinct knots such that $2$-handle additions along them with framing $n$ yield the same $4$-manifold. As a corollary, we obtain the affirmative answer to a problem in Kirby’s list (Problem 3.6 (D)). We also explain a relation between our result and the cabling conjecture. This is a joint work with Tetsuya Abe.

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講 演 者: 佐藤 光樹 (東京学芸大学)
タ イ ト ル: Non-orientable genus of a knot in punctured Spin 4-manifolds

For a closed 4-manifold $M$ and a knot $K$ in the boundary of punctured $M$, we define $\gamma_M^0(K)$ to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured $M$ with boundary $K$. Note that $\gamma^0_{S^4}$ is equal to the non-orientable 4-ball genus and hence $\gamma^0_M$ is generalization of the non-orientable 4-ball genus. While it is very likely that for given $M$, $\gamma^0_M$ has no upper bound, it is difficult to show it. In fact, even in the case of $\gamma^0_{S^4}$, its non-boundedness was shown for the first time by Batson in 2012. In this talk, we show that for any Spin 4-manifold $M$, $\gamma^0_M$ has no upper bound.

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講 演 者: 岩切 雅英 (佐賀大学)
タ イ ト ル: Unknoting numbers for handlebody-knots and Alexander
quandle colorings

A crossing change of a handlebody-knot is that of a spatial graph representing it. We see that any handlebody-knot can be deformed into trivial one by some crossing changes. So we define the unknotting numbers for handlebody-knots. In the case classical knots, which are considered as genus one handlebody-knots, Clark, Elhamdadi, Saito and Yeatman gave lower bounds of the Nakanishi indices by the numbers of some finite Alexander quandle colorings, and hence they also gave lower bounds of the unknotting numbers. In this talk, we give lower bounds of the unknotting numbers for handlebody-knots with any genus by the numbers of some finite Alexander quandle colorings of type at most 3.

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講 演 者: 屋代 司 (Sultan Qaboos University)
タ イ ト ル: Lower decker sets and triple points for surface-knots

A surface-knot is a closed oriented surface embedded in 4-space. A surface diagram of a surface-knot is the projected image in 3-space under the orthogonal projection with crossing information. The pre-image of multiple point sets of a surface diagram is called a double decker set that is the union of lower and upper decker sets. A crossing point in a lower decker set corresponds to a triple point in the diagram. If a diagram is coloured by a quandle, we can define degenerate triple points. In this talk we estimate the number of non-degenerate triple points in a coloured diagram corresponding to a connected component of the lower decker set.

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講 演 者: Amal Al Kharusi (Sultan Qaboos University)
タ イ ト ル: Unknotting operation and independent components of lower
decker set

This talk is divided into two main parts. In the first part, a generalization of the notion of crossing change operation of knot diagrams to surface-knot diagrams will be discussed. The second part involves a method for detecting prime surface-knots by using the colourings. In addition, the notion of independent components of lower decker set will be demonstrated.

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講 演 者: Daniele Zuddas (Korea Institute for Advanced Study)
タ イ ト ル: An equivalence theorem for Lefschetz fibrations over the disk

In this talk we present the moves for (achiral) Lefschetz fibrations over the disk, that give a complete interpretation of Kirby calculus for 4-manifolds in the setting of Lefschetz fibrations.
We discuss also an application to open book decompositions of 3-manifolds.

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講 演 者: 藤井 道彦 (京都大学)
タ イ ト ル: On minimal crossing expressions of braids

First, we give a usual projection of braid onto the two dimensional Euclidean space.
Then we consider the problem how a minimal crossing braid is obtained from the projection.

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講 演 者: Fedor Duzhin (Nanyang Technological University)
タ イ ト ル: Brunnian braids and Brunnian links

A Brunnian link is a link that becomes trivial after removing any of its components. Similarly, a Brunnian braid is a braid that becomes trivial after removing any of its strands. According to Alexander's theorem, any link can be obtained as a closure of some braid. Obviously, the closure of a Brunnian braid is a Brunnian link. However, some not every Brunnian link is the closure of a Brunnian braid. In this talks we'll discuss some ways of constructing Brunnian links that cannot be obtained as a closure of a Brunnian braid.

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講 演 者: 鎌田 直子 (名古屋市立大学)
タ イ ト ル: Writhes of a twisted knot derived from weighted index diagrams

The odd writhe is a numerical invariant of a virtual knot which is defined by L. Kauffman. Y. Im, K. Lee and Y. Lee and A. Henrich introduced a polynomial invariant of a virtual knot, called the index polynomial. S. Satoh and K. Taniguchi introduced a series of numerical invariants of a virtual knot, called n-writhes, which induce the odd writhe and are related with the index polynomial. In this talk, we discuss a generalization of them to a twisted knot by use of weighted index diagrams.

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講 演 者: 浮田 卓也 (東京工業大学)
タ イ ト ル: Genus zero PALF structures on the Akbulut cork and exotic
pairs

Loi and Piergallini proved that every compact Stein surface admits a PALF (positive allowable Lefschetz fibration over a 2-disk with bounded fibers). Akbulut and Yasui introduced cork twist to construct various families of arbitrary many compact Stein surfaces which are mutually homeomorphic but not diffeomorphic (i.e. exotic pairs). In this talk, we construct genus zero PALF on the Akbulut cork and exotic pairs.

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講 演 者: 田中 利史 (岐阜大学)
タ イ ト ル: On the maximal Thurston-Bennequin number of knots and
links in a spatial graph

A spatial graph is said to be mTB-realizable if it is ambient isotopic to a Legendrian graph such that all its cycles realize their maximal Thurston-Bennequin numbers. In this talk, I will give an example of an infinite family of spatial embeddings of the complete graph on 4 vertices that are mTB-realizable and show that if a spatial graph contains a completely splittable link as the union of all its cycles, then it is mTB-realizable. I will also show that if a finite graph G contains two cycles that have no common edges and vertices, then there is a spatial embedding of G such that it is not mTB-realizable.

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講 演 者: 門田 直之 (大阪電気通信大学)
タ イ ト ル: Non-holomorphic Lefschetz fibrations with (-1)-sections

The notion of Lefschetz fibrations in two-dimentional complex geometry (i.e., holomorphic Lefschetz fibrations) was generalized by Moishezon to four-dimensional topology. It is then natural to ask how far Lefschetz fibrations in four-dimensional topology are from holomorphic ones. To answer this question, various kinds of examples of non-holomorphic Lefschetz fibrations have been constructed. In particular, from Donaldson's theorem, it follows that there are infinitely many non-holomorphic Lefschetz fibrations with (-1)-sections. In this talk, we construct explicit examples of non-holomorphic Lefschetz fibrations with (-1)-sections without Donaldson's theorem. This is a joint work with Noriyuki Hamada (The university of Tokyo) and Ryoma Kobayashi (Tokyo university of science).

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講 演 者: 河村 建吾 (大阪市立大学)
タ イ ト ル: On the clasp number of a knot

Every knot in $S^{3}$ bounds a singular disk in $S^{3}$ whose singular set consists of only clasp singularities. The clasp number $c(K)$ of a knot $K$ is the minimal number of clasp singularities among all such singular disks bounded by $K$. In this talk, we determine the clasp number of a certain knot by using a characterization of the Alexander module, and we investigate the clasp numbers of $2$-bridge knots.
This is a joint work with T. Kadokami.

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講 演 者: 植木 潤 (九州大学)
タ イ ト ル: On the universal deformations of SL(2)-representations of
2-bridge knot groups

Joint work with M. Morishita, Y. Takakura and Y. Terashima.
In this talk, we first review Alexander-Fox theory and Iwasawa theory as the moduli theories of 1-dim. representations of knot/prime groups $G$. Then, we study the analogue story of prime groups, Hida-Mazur’s theory of 2-dim. representations, for knot groups:
(1) we overview the deformation theory of SL(2)-representation,
(2) state the relation between character variety and universal deformation ring, and
(3) construct the universal deformation of Riley's standard SL(2)-representation of 2-bridge knot group.
It should be remarked that In the dictionary of analogy in Arithmetic Topology, knots correspond to primes, and GL(2) Alexander polynomials $A(r,s)$ correspond to $p$-adic L-functions $L(r,s)$, where the parameters $s$ and $r$ come from GL(1) and SL(2) theories respectively.

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最終更新日: 2014年11月26日
(C)大阪市大数学教室