Friday Seminar on Knot Theory(2007年度)

組織委員 田中 利史

日時 2008年3月28日(金)15:00~16:00
講演者(所属) 川見 将広(OCAMI)
タイトル Mod4 quadratic forms and diffeomorphisms on non-orientable surfaces
場所 数学 第3セミナー室(3153)
アブストラクト We investigate Mod4 quadratic forms on the $\mathbb{Z}_{2}$-coefficient first homology group of a non-orientable surface. In the case of the genus of a surface is small, we can enumerate them up to isomorphims easily and they are few. In this talk, we study the self-diffeomorphims on a non-orientable surface which leave a given Mod4-quadratic form invariant.
日時 2008年2月29日(金)16:00~17:00
講演者(所属) 秋吉 宏尚(OCAMI)
タイトル Side parameter for the punctured torus groups
場所 数学 第3セミナー室(3153)
アブストラクト The side parameter for the quasifuchsian punctured torus groups is defined by Jorgensen in his famous unfinished paper on the combinatorial structures of the Ford domains of such groups. The side parameter is a complete invariant on the quasifuchsian space, and is extended to an invariant on the closure of the space. The main results of this talk are the following:
1. The extended side parameter is also a complete invariant.
2. The composition of the inverse of the end invariant map and the side parameter map is a self-homeomorphism on the common target space of the invariants.
日時 2008年2月22日(金)16:30~17:30
講演者(所属) 中村 拓司(大阪電気通信大学)
タイトル On knots of Delta unknotting number one from a view of the positivity for knots
場所 数学 第3セミナー室(3153)
アブストラクト A Delta unknotting number for a knot is the minimal number of ``Delta moves" needed to create the unknot. A knot is said to be positive if it has a diagram whose all crossings are positive. In this talk, we show that a Delta unknotting number one positive knot is the right-handed trefoil knot. We also observe almost positive knots of Delta unknotting number one.
日時 2008年2月22日(金)15:30~16:30
講演者(所属) Alexander Stoimenow(OCAMI, COE fellow)
タイトル Vassiliev invariants, Seifert matrix, and hyperbolic volume of knots
場所 数学 第3セミナー室(3153)
アブストラクト Given any knot K, we construct hyperbolic knots with arbitrarily large volume, with the same Seifert matrix and the same Vassiliev invariants of a bounded degree as K. An additional feature of our knots is that they have bounded free genus.
日時 2008年2月8日(金)16:00~17:00
講演者(所属) 森内 博正(OCAMI)
タイトル Classifications of theta-curves and handcuff graphs
場所 数学 第3セミナー室(3153)
アブストラクト We have enumerated all the theta-curves and handcuff graphs with up to seven crossings. To make a table of them, we need some spatial graph invariants.
In 1989, S. Yamada defined the topological invariant of spatial graphs known as the Yamada polynomial. In this talk, we compute the Yamada polynomial of some spatial graphs, and mention some properties of the Yamada polynomial.
日時 2008年2月1日(金)16:00~17:00
講演者(所属) Thomas Mattman(California State University)
タイトル Boundary Slope Diameter and Crossing Number of 2-Bridge Knots
場所 数学 第3セミナー室(3153)
アブストラクト In joint work with Maybrun and Robinson, we prove that for 2-bridge knots, the diameter of the set of boundary slopes is twice the crossing number. After some topological preliminaries, the argument quickly becomes combinatorial and is based on comparing various continued fraction representations of the same fraction.
日時 2007年12月14日(金)13:30~14:30
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 The sharp-unknotting number of a torus knot
場所 数学 第3セミナー室(3153)
アブストラクト The sharp-unknotting number was introduced by Hitoshi Murakami. He estimates for the sharp-unknotting number from below using the signature, Arf invariant, homology invariant from a cyclic covering. We give an estimation for the sharp-unknotting numbers of torus knots, which determines for some cases.
日時 2007年11月30日(金)15:30~16:30
講演者(所属) 田山 育男(OCAMI)
タイトル Enumerating 3-manifolds with lengths up to 9 by a canonical order
(joint work with Akio Kawauchi (Osaka City University))
場所 数学 第3セミナー室(3153)
アブストラクト This is a joint work with A. Kawauchi. A well-order was introduced on the set of links by A. Kawauchi. This well-order also naturally induces a well-order on the set of prime link exteriors and eventually induces a well-order on the set of closed connected orientable $3$-manifolds. With respect to this order, we enumerated the prime links with lengths up to 10 and the prime link exteriors with lengths up to 9. In this talk, we show a list (with several pending manifolds) of the enumeration of $3-$manifolds with lengths up to 9 by using the enumeration of the prime link exteriors.
日時 2007年11月30日(金)14:00~15:00
講演者(所属) Daniel Moskovich
タイトル Two Surgery Presentations for Dihedral Covering Spaces
(joint work with Andrew Kricker (Nanyang Technological University))
場所 数学 第3セミナー室(3153)
アブストラクト This is joint work with A. Kricker. We present two different but related procedures for obtaining surgery presentations of dihedral covering spaces of $S3$ branched along knots. In particular we show that for any odd prime p there are exactly p equivalence classes of p-coloured knots modulo surgery by unit-framed (unknotted) components representing elements in the kernel of the p-colouring.
日時 2007年11月16日(金)16:00~17:00
講演者(所属) 伊藤 昇(早稲田大学)
タイトル Invariants via word for curves
場所 数学 第3セミナー室(3153)
アブストラクト We construct an infinite sequence of invariants for curves in surfaces by using word theory that V. Turaev introduced. For plane closed curves, we add some extra terms, e.g. the rotation number. From these modified invariants, we get the Arnold's basic invariants and some other invariants. We also express how these invariants classify plane curves.
日時 2007年11月9日(金)16:00~17:00
講演者(所属) Yo'av Rieck(University of Arkansas)
タイトル On the Heegaard genus of knot exteriors
(joint with Tsuyoshi Kobayashi (Nara Women's University))
場所 数学 第3セミナー室(3153)
アブストラクト We will survey some of the authors' results about the behavior of Heegaard genus of knot exteriors under connected sum operation. As our main result we will prove that given integers $g_i > 1 (i=1,...,n)$, there exist knots $K_i$ in $S3$ so that:

1) $g(E(K_i)) =g_i$, and:
2) $g(E(K_1\sharp ...\sharp K_n)) = g(E(K_1)) +...+ g(E(K_n))$.

This proves the existence of counterexamples to Morimoto's Conjecture.
日時 2007年10月12日(金)15:00~16:00
講演者(所属) 門上 晃久(OCAMI)
タイトル Calculating the Casson-Walker invariants of cyclic branched coverings over knots yielding lens spaces
(a joint work with Yasuyoshi Tsutsumi (Oshima National College of Maritime Technology)
and Yukihiro Tsutsumi (Sophia University))
場所 数学 第3セミナー室(3153)
アブストラクト We exhibit a technique for calculating concretely the Casson-Walker invariants of cyclic branched coverings over knots yielding lens spaces on special cases. By using it, we consider problems such as detecting problem of branched covering spaces, and lens surgery problem in a homology 3-spheres.
日時 2007年10月12日(金)14:00~15:00
講演者(所属) 福本 善洋(鳥取環境大学)
タイトル Homology spin cobordism problem of plumbed 3-manifolds and the cup product structures
場所 数学 第3セミナー室(3153)
アブストラクト In this talk, we introduce two approaches to give necessary conditions for abstract isomorphisms on homology of two plumbed 3-manifolds to be realized by homology spin cobordisms geometrically. In the first approach we use a V-manifold version of the Furuta-Kametani 10/8-inequality for closed spin 4-manifolds to obtain a necessary condition in terms of an integral lift of the Rochlin invariant and the quadruple cup product structure. In particular, we formally calculate cup products by using the data of abstract isomorphism on homology between plumbed 3-manifolds. In the second approach we use the associativity of cup products. In fact, the formal calculations of cup products may fail to satisfy the associativity law. Motivated by this, we introduce a certain triple product to give other necessary conditions.
日時 2007年7月13日(金)16:00~17:00
講演者(所属) 佐藤 隆夫(大阪大学大学院理学研究科)
タイトル Twisted homology groups of the automorphism group of a free group
場所 数学 第3セミナー室(3153)
アブストラクト In this talk, we compute twisted first and second homology groups of the automorphism group of a free group with coefficients in the abelianization of a free group and its dual group, using a presentation sm group of a free group due to Gersten.
日時 2007年7月6日(金)16:00~17:00
講演者(所属) 田中 利史(OCAMI)
タイトル Maximal Thurston-Bennequin numbers and Rasmussen invariants of doubled knots
場所 数学 第3セミナー室(3153)
アブストラクト Maximal Thurston-Bennequin number is a knot invariant from contact geometry. It is well-known that the invariant is strictly less than the minimum v-degree of the Kauffman polynomial in the framing variable v. Recently, we showed that the Kauffman bound is sharp for any positive knot and any alternating knot. However, it is known to be not sharp for many other knots in general. In this talk, we confirm that the Kauffman bound is sharp for any double of a knot if the bound is sharp for the knot. On the other hand, it is also known that the maximal Thurston-Bennequin number is strictly less than Rasmussen's s-invariant. We also discuss the s-invariant bound for the maximal Thurston-Bennequin numbers of doubled knots.
日時 2007年6月29日(金)16:00~17:00
講演者(所属) 牛島 顕(金沢大学大学院自然科学研究科)
タイトル Hyperbolic spatial graphs coming from strongly invertible knots
(joint work with Kazuhiro Ichihara (Nara University of Education))
場所 数学 第3セミナー室(3153)
アブストラクト There is a way, called "rational-fold cyclic branched covering," to construct spatial graphs from an invertible knot. In this talk we will give a condition, which is expected to be a necessary and sufficient one, for strongly invertible knots in the 3-sphere to yield hyperbolic spatial graphs. We will also see that strongly invertible simple knots and tunnel number one knots satisfy it so that we can have infinitely many hyperbolic spatial graphs in the 3-sphere.
日時 2007年6月15日(金)16:00~17:00
講演者(所属) 石井 敦(京都大学数理解析研究所)
タイトル The skein index for link invariants
場所 数学 第3セミナー室(3153)
アブストラクト We introduce the skein index, which is an integer-valued index t is used to compare link invariants and to find a skein relation. We give the complete list for link invariants of skein index less than or equal to two, and discuss the skein index of an operator invariant.
日時 2007年6月8日(金)16:00~17:00
講演者(所属) 佐藤 進(神戸大学大学院理学研究科)
タイトル The sheet numbers of 2-knots
場所 数学 第3セミナー室(3153)
アブストラクト A $2$-knot is an embedded $2$-sphere in $4$-space, and its diagram is a projection image of the $2$-knot into $3$-space together with crossing information. Such a diagram is regarded as a disjoint union of compact connected surfaces each of which is called a sheet. The sheet number of a $2$-knot is defined as the minimal number of sheets for all possible diagrams of the $2$-knot. The notion of the sheet number is analogous to the crossing number of a classical knot in $3$-space. In this talk, we give a lower bound of the sheet number in several ways (Fox colorings, fundamental quandles, and cocycle invariants), and determine the sheet numbers of the $2$-, $3$-twist-spun trefoils, and some ribbon $2$-knots.
日時 2007年6月1日(金)16:00~17:00
講演者(所属) 岩切 雅英(OCAMI)
タイトル Quandle cocycle invariants of charts with six white vertices
場所 数学 第3セミナー室(3153)
アブストラクト By $S_p(\Gamma)$, we denote the quandle cocycle invariant of a chart $\Gamma$ associated with Mochizuki's $3$-cocycle of the dihedral quandle of order $p$. In this talk, we prove that there is no chart $\Gamma$ with six white vertices such that $S_p(\Gamma)$ is non-trivial for some odd prime integer $p$ but $3$. This gives lower bounds of the w-indices of some surface links. We also consider charts with six white vertices such that $S_3(\Gamma)$ is non-trivial.
日時 2007年5月11日(金)16:00~17:00
講演者(所属) 丹下 基生(大阪大学大学院理学研究科)
タイトル タイトな接触構造とレンズ手術について
場所 数学 第3セミナー室(3153)
アブストラクト レンズ空間を生む3次元球面の中の結び目のよく知られている制限とそれを満たす結び目を紹介する。
その制限を一般のホモロジー球面の場合に拡張し、そのホモロジー球面がL-spaceである場合にそれを満たす結び目の系列について述べる。また3次元の接触構造を使って正の手術でレンズ空間を得ることができないホモロジー球面が存在することを示す。
日時 2007年4月27日(金)16:00~17:00
講演者(所属) 平澤 万希子(OCAMI)
タイトル A classification of links up to 5-move equivalence
場所 数学 第3セミナー室(3153)
アブストラクト Montesinous and Nakanishi conjectured that 3-move and 4-move are unknotting operations around 1980. In 2002, Dabkowski and Przytycki proved that 3-move is not an unknotting operation via argument of Burnside group, but 4-move conjecture remained unsettled. In this talk, we consider 5-move equivalence relation, and classify rational links and links up to 9-crossings.
日時 2007年4月20日(金)16:00~17:00
講演者(所属) 山本 亮介(OCAMI)
タイトル Complexity of open book decompositions via arc complex
(Joint work with Toshio Saito (Nara Women's University))
場所 数学 第3セミナー室(3153)
アブストラクト Based on Hempel's distance of a Heegaard splitting, we define a certain kind of complexity of an open book decomposition, called a translation distance, by using the arc complex of its fiber surface. We then show that an open book decomposition is of translation distance at most two if it is split into "simpler" open book decompositions and at most three if it admits a Stallings twist on it.
日時 2007年4月13日(金)16:00~17:00
講演者(所属) 安井 弘一(大阪大学大学院理学研究科)
タイトル Small exotic rational surfaces without 1- and 3-handles
場所 数学 第3セミナー室(3153)
アブストラクト Jongil Park \textit{et al.} constructed exotic $\mathbf{CP}^2\# n\overline{\mathbf{CP}}^2\,(5\leq n\leq 8)$ by using rational blow-downs and elliptic fibrations. In this talk we give another construction by using rational blow-downs and Kirby calculus. We also prove our manifolds admit handle decompositions without $1$- and $3$-handles, in the case $7\leq n\leq 9$. Note that every exotic $\mathbf{CP}^2$, if it exists, has at least either a $1$- or $3$-handle in each handle decompositon of it.
最終更新日: 2008年3月21日