## Friday Seminar on Knot Theory（2019年度）

6月21日　講演者：木村直記（早稲田大学）
6月28日　お休み

7月 5日　講演者：小鳥居祐香（理化学研究所／大阪大学）

 日時 2019月7月5日（金）16：00～17：00 講演者（所属） 小鳥居祐香（理化学研究所／大阪大学） タイトル On Levine’s classification of link-homotopy classes of 4-component links 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト Two links are link-homotopic if they are transformed to each other by a sequence of self-crossing changes and ambient isotopies. Milnor introduced this notion and classified 2- and 3-component links up to link-homotopy by Milnor invariants. Levine classified the set of the link-homotopy classes of 4-component links and gave some subsets of them which were classified by invariants as a corollary. In this talk, we modify the results by using Habiro's clasper theory. The new classification allows us schematic treatment of the link-homotopy classes of 4-component links which has symmetry with respect to the components. We also give some new subsets of them which are classified by invariants. This is joint work with Atsuhiko Mizusawa (Waseda University).
 日時 2019月6月21日（金）16：00～17：00 講演者（所属） 木村 直記 (早稲田大学) タイトル Dijkgraaf-Witten invariants of cusped hyperbolic 3-manifolds 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト The Dijkgraaf-Witten invariant is a topological invariant defined for closed oriented 3-manifolds in terms of a finite group and its 3-cocycle. Dijkgraaf and Witten gave a combinatorial construction of the invariant by using a triangulation. Wakui showed the invariance of this combinatorial construction. In this talk, we introduce a generalization of the Dijkgraaf-Witten invariant for cusped oriented 3-manifolds by using an ideal triangulation, and we calculate the generalized invariants for some cusped hyperbolic 3-manifolds.
 日時 2019月6月14日（金）16：00～17：00 講演者（所属） 福田瑞季（東京学芸大学） タイトル Gluck twist on branched twist spins 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト A branched twist spin is a smoothly embedded two sphere in the four sphere and it is defined as the set of non-free orbits of a circle action on the four sphere. Gluck showed that the set of isotopy classes of diffeomorphisms on $S^1 \times S^2$ is isomorphic to $\mathbb{Z}_2$, and an operation of removing a neighborhood of a 2-knot from the four sphere and regluing it by the generator of $\mathbb{Z}_2$ is called a Gluck twist. It is known by Pao that the Gluck twist along a branched twist spin does not change the four sphere. In this talk, we give an another proof of Pao’s result by using a decomposition of $S^4$ associated with the circle action, and we show that the set of branched twist spins does not change by the Gluck twist.
 日時 2019月5月17日（金）16：00～17：00 講演者（所属） 軽尾 浩晃 (京都大学数理解析研究所) タイトル The reduced Dijkgraaf–Witten invariant of twist knots in the Bloch group of $\Bbb{F}_p$ 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト For a closed 3-manifold $M$, a group $G$, a 3-cocycle $\alpha$ of $G$, and a representation $\rho \colon \pi_1(M) \to G$, the Dijkgraaf–Witten invariant is defined to be $\rho^\ast \alpha [M]$, where $[M]$ is the fundamental class of $M$, and $\rho^\ast \alpha$ is the pull-back of $\alpha$ by $\rho$. We consider an equivalent invariant $\rho_\ast [M] \in H_3(G)$, and we also regard it as the Dijkgraaf–Witten invariant. In 2004, Neumann described the hyperbolic volume and Chern–Simons invariant of $M$ in terms of the image of the Dijkgraaf–Witten invariant for $G={\rm SL}_2 \Bbb{C}$ by the Bloch–Wigner map $H_3(M)\to \mathcal{B}(\Bbb{C})$, where $\mathcal{B}(\Bbb{C})$ is the Bloch group of $\Bbb{C}$. Further, in 2013, Hutchinson gave a construction of the Bloch–Wigner map $H_3({\rm SL}_2 \Bbb{F}_p)\to \mathcal{B} (\Bbb{F}_p)$, where $p$ is a prime, and $\Bbb{F}_p$ is the finite field of order $p$. In this talk, I calculate the reduced Dijkgraaf–Witten invariant of the complement of twist knots, where the reduced Dijkgraaf–Witten invariant is the image of the Dijkgraaf–Witten invariant for ${\rm SL}_2 \Bbb{F}_p$ by the Bloch–Wigner map $H_3({\rm SL}_2\Bbb{F}_p) \to \mathcal{B} (\Bbb{F}_p)$.
 日時 2019月5月10日（金）16：00～17：00 講演者（所属） 阿蘇 愛理 (首都大学東京) タイトル Twisted Alexander polynomials of tunnel number one Montesinos knots 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト The twisted Alexander polynomial was introduced in 1990's as a generalization of the Alexander polynomial, which is one of the classical invariants of knots determined by the fundamental groups of the complement of knots (i.e. knot groups). Since the twisted Alexander polynomial of a knot is determined by the knot group and their representation, it has more information than the classical Alexander polynomial. For example, Kinoshita-Terasaka and Conway's 11 crossing knots, which are not distinguished by their Alexander polynomials, are distinguished by their twisted Alexander polynomials. The tunnel number of a knot $K$ is the minimal number of mutually disjoint arcs $\{ \tau_i \}$ in $S^3\setminus K$ such that the component of an open regular neighborhood of $K \cup (\cup \tau_i)$ is a handlebody. In this talk, we calculate the twisted Alexander polynomials of tunnel number one Montesinos knots associated to their $SL_2(\mathbb{C})$ representations.
 日時 2019月4月19日（金）16：00～17：00 講演者（所属） 河村建吾（大阪市立大学数学研究所） タイトル A simple calculation of the Arf invariant of a proper link 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト An oriented link $L$ is said to be proper if the linking number $\mathrm{lk}(K,L \setminus K)$ is even for each component $K$ in $L$. The Arf invariant $\mathrm{Arf}(L)\in\{0,1\}$ of a link $L$ is defined only when $L$ is a proper link. It is known that there are several ways to calculate $\mathrm{Arf}(L)$, e.g. using Seifert forms, the polynomial invariants, local moves, and 4-dimensional techniques. However, it is not easy to use such methods for an arbitrarily given proper link. In this talk, we introduce a simple way to calculate the Arf invariant for such a proper link.
 日時 2019月4月12日（金）16：00～17：00 講演者（所属） 清水理佳（群馬工業高等専門学校） タイトル The warping sum of knots 場所 理学部 Ｆ棟 415号室（中講究室） アブストラクト An oriented knot diagram is said to be monotone if one can travel along the diagram so that one meets each crossing as an over-crossing first starting at a point on the diagram. The warping degree of an oriented knot diagram is the minimum number of crossing-changes which are required to obtain a monotone diagram from the knot diagram. For an unoriented knot diagram, the warping sum is the value of the sum of the warping degrees with both orientations. We define the warping sum of an unoriented knot to be the minimal value of the warping sum for all minimal-crossing diagrams of the knot. It has been shown that the warping sum is less than or equal to the crossing number minus one for any knot, and the equality holds if and only if the knot is prime and alternating. We also define a knot invariant, the reduced warping sum of a knot, to be the minimal value of the warping sum for all diagrams of the knot. In this talk, we determine knots with warping sum and reduced warping sum three or less. This is a joint work with Slavik Jablan.