理学部数学科 OCAMI 今年度

Friday Seminar on Knot Theory（2015年度）

The concept of "bridge" is widely known to knot theorists, but in some different ways. The difference turns out substantial when they discuss isotopies of it. In this talk, I will compare the notions of bridge position, bridge decomposition and bridge sphere, and survey the structures formed by the isotopy classes of them. This is partially joint work with Y. Jang, T. Kobayashi and M. Ozawa.

Confoliation is an intermediary notion between contact structure and foliation. One of the main theme of confoliation theory is to deform a contact structure to a folaition. We can deform a given contact structure $D$ on a closed 3-manifold into (the tangent bundle of) a foliation $F$ through a family of contact structures. Consider a transverse knot $K$ of $D$ and deform it through a family of transverse knots into a transverse knot of $F$. Of course, in general, there is no such deformations. In the happy case where any transverse knot of $D$ can be deformed into that of $F$, we say that the convergence of $D$ into $F$ is good. As for a good convergence, $D$ is tight if and only if $F$ satisfies the relative Thurston inequality. We sketch the proof of good convergence in a certain case. This talk is partially based on a collaboration with Yoshihiko Mitsumatsu.

A quandle is a set equipped with a binary operation satisfying certain axioms derived from Reidemeister moves. J.S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito introduced quandle cohomology theory and quandle cocycle invariants for oriented links and surface-links. J.S. Carter, S. Kamada and M. Saito defined shadow quandle cocycle invariants for oriented links and surface-links, and S. Kamada and K. Oshiro defined symmetric quandle cocycle invariants for unoriented surface-links using symmetric quandle homology theory. These quandle cocycle invariants for surface-links are defined by using broken surface diagrams. In this talk, I'd like to describe these quandle cocycle invariants via marked graph diagrams and introduce shadow biquandle cocycle invariants for oriented surface-links. This is a joint work with S. Kamada and S.Y. Lee.

First, we will give formulas to calculate the Homfly polynomial of knots formed by 3-tangles. After that, we will construct families of non-alternating knots and give explicit formulas to calculate the Alexander polynomial of them. The knots in these families are prime and of alternation number one. Furthermore, they are hyperbolic except for the only two torus knots. The families contain the first non-alternating knots: $8_{19}$, $8_{20}$, $8_{21}$.

In this talk, we review the construction of a polynomial invariant of graphs in 3-space. As an application, we give some estimates for the tunnel number and the generalized bridge number of bouquets.

A knotted torus or Klein bottle in Euclidian $4$-space is called a $T^2$-knot or Kb-knot, respectively. It is known that any ribbon $T^2$-knot is presented by a welded knot. In this talk, we will introduce the notion of noded knots and prove that any ribbon Kb-knot is presented by some noded knot. We also study several properties of a noded knot presentation. In particular, we discuss a Kb-knot presented by a noded knot obtained from a classical knot by replacing a classical crossing with a welded one.

日時	：	10月9日（金）16：30～17：30
講演者	：	河村建吾 (大阪市立大学）
タイトル	：	Surface-knots with self-intersections in the 4-space
場所	：	理学部 Ｆ棟 415号室 (中講究室）

Ribbon singularity and clasp singularity are typical types of singularities appearing in immersions of surfaces into the 3-space. A knot is called ribbon if it appears as the boundary of an immersed 2-disk with ribbon singularities. A 2-knot or a surface-knot is called ribbon if it bounds an immersed 3-disk or handlebody with ribbon singularities. In this talk, we introduce the notion of a ribbon-clasp surface-knot, which appears as the boundary of an immersed 3-disk or handlebody with ribbon singularities and clasp singularites.

We consider a random link, which is defined as the closure of a braid obtained from a random walk on the braid group. For such a random link, the expected value for the number of components was calculated by Jiming Ma. In this talk, we determine the most expected number of components for a random link, and further, consider the most expected partition of the number of strings for a random braid. This is based on joint work with Ken-ichi Yoshida (Nihon University).

Cork is a compact Stein surface which gives rise to exotic pairs of 4-manifolds. We find infinitely many corks with shadow complexity one among the 4-manifolds constructed from contractible special polyhedra having one true vertex by using the notion of Turaev's shadow. We also show that there are just two types of polyhedra which are shadows of corks with shadow complexity one.

We consider random walk on the mapping class group to generate "random" mapping classes. In the first half of this talk I will summarize basic properties and important known facts about random walk on the mapping class group. Then I will introduce the notion called fibered commensurability and discuss it for random mapping classes.

Loi and Piergallini showed that any compact Stein surface is the total space of a simple branched covering of a 4-ball whose branch set is a positive braided surface. They also showed that the opposite is true. Unfortunately, although the fact is well-known, little is known about how Stein structures behave towards positive braided surfaces. In this talk, we give an infinite family of positive braided surfaces as branch sets of simple branched coverings whose total spaces are all diffeomorphic but admit mutually different Stein structures.

日時	：	6月26日（金）16：00～17：00
講演者	：	安井 弘一（広島大学）
タイトル	：	Corks, exotic 4-manifolds and knot concordance
場所	：	理学部 Ｆ棟 415号室 (中講究室）

We give a method for producing framed knots which represent homeomorphic but non-diffeomorphic (Stein) 4-manifolds, using corks and satellite maps. To obtain the method, we introduce a new description of cork twists. As an application, we construct knots with the same 0-surgery which are not concordant for any orientations. This disproves the Akbulut-Kirby conjecture given in 1978.

This is a joint research with Scott Carter, Atsushi Ishii and Kokoro Tanaka. A quandle is a set that has a binary operation satisfying three conditions corresponding to the Reidemeister moves. A quandle that is a union of groups with the operation restricting to conjugation on each group component is called a multiple conjugation quandle. In this definition, compatibilities between the group and quandle operations are imposed which are motivated by considerations on colorings of handlebody-links. In this talk, a homology theory for multiple conjugation quandles is presented that unifies group and quandle homology theories. Algebraic aspects, such as extensions, are discussed, and degenerate subcomplexes are defined in relation to simplicial decompositions of prismatic (products of simplices) complexes. Cocycle invariants are defined for handlebody-links using this homology theory.

We explain a description of quantum $sl_2$ invariants (loop expansion, colored Alexander invariant) of knots in terms of homological braid group representations, which helps us to understand topological content of quantum $sl_2$ invariants.

参考文献： arXiv:1411.5418   arXiv:1505.02841

Let $X_n$ be a 4-manifold obtained by attaching two trefoils with linking number $1$ and with the framings $(0,n)$. Y. Matsumoto asked in the Kirby's problems list whether two generators in $H_2(X_0)$ can be realized by the embedded edge of two spheres. We discuss when the boundary $M_n$ of $X_n$ bounds a contractible 4-manifold by using Heegaard Floer homology. $M_n$ is 1-surgery of the n-twisted Whitehead double $K_n$ of the trefoil, and we will determine the 4-ball genus of $K_n$ by using the obstruction by Owens and Strle.

The Alexander polynomial is an effective knot invariant untill now. Levine and Rolfsen introdoced a surgical view of Alexander invariants. In this talk, the speaker will talk on the surgical view and its applications: unknotting number and knot adjacency.

Beside Artin's standard "combing normal form" and Garside-Adjan-Morton-Thurston equally well-known "greedy normal form", we develop another natural and simple normal form of braids based on the embedding of (n-1)-strand braids into n-strand braids. This approach is specially well adapted for analyzing the canonical ordering of positive braids and it leads to paradoxically long sequences and, from there (joint work with L.Carlucci and A.Weiermann), to unprovability statements for certain games involving braids.

In knot theory, knot diagrams play an important role to study and classify knots. For the crossing points of an oriented knot diagram, we have three points of view: over/under crossing, left/right crossing and positive/negative crossing. In 2012, Higa, Nakanishi, Satoh and Yamamoto defined an OU sequence for a knot diagram, where the OU sequence is obtained from crossing information by reading a sequence of over/under crossing points along the orientation direction of a knot. They mainly studied sequences which are realized by diagrams of the trefoil knot. In this talk, we will focus on left/right crossing and positive/negative crossing for a knot diagram and study the cyclic words obtained from the crossing sign. In the case of focusing on left/right crossing information, this information does not reflect any over/under crossing information. Therefore we treat a spherical closed curve instead of a knot diagram.

We calculate the twisted Alexander ideals for spatial graphs, handlebody-knots, and surface-links. For spatial graphs, we calculate the invariants of Suzuki's theta-curves and show that the invariants are nontrivial for Suzuki's theta-curves whose Alexander ideals are trivial. For handlebody-knots, we give a remark on abelianizations and calculate the invariant of the handlebody-knots up to six crossings. For surface-links, we correct Yoshikawa's table and calculate the invariants of the surface-links in the table. This is a joint work with Atsushi Ishii (University of Tsukuba) and Kanako Oshiro (Sophia University).

A finite set of two-dimensional manifolds embedded in three-dimensional Euclidean space is called an $\textit{arrangement}$ $\textit{of}$ $\textit{surfaces}$. A link $L$ is said to be $\textit{arrangeable}$ on an arrangement $\mathcal{F}$ of surfaces if there exists a link $L'$ which is ambient isotopic to $L$ such that each component of $L'$ is contained in a surface belonging to $\mathcal{F}$. We consider the following problems. (1) Given an arrangement of surfaces, determine links which can be arrangeable on it. (2) Given a link, determine arrangements of surfaces on which the link is arrangeable. I will talk about partial answers to the problems.

It is known that every knot bounds a singular disk whose singular set consists of only clasp singularities. Such a singular disk is called a clasp disk. The clasp number of a knot is the minimum number of clasp singularities among all clasp disks of the knot. The $\Gamma$-polynomial is the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials. I will talk about a characterization of the $\Gamma$-polynomials of knots with the clasp numbers at most two.

A classical Gluck's theorem states that there exist at most two inequivalent 2-knots with diffemorphic exteriors. In this talk, we construct infinitely many ribbon disks with the same exterior. First, we give a sufficient condition for a given slice disk to be ribbon. Next, we construct infinitely many slice disks with the same exterior, and prove that these are ribbon. This is a joint work with Motoo Tange. If time permits, we prove that these ribbon disks are mutually distinct by the (overtwisted) contact structures in the 3-sphere.