Friday Seminar on Knot Theory

Friday Seminar on Knot Theory

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Organizer:   Hirotaka Akiyoshi   (

Upcoming speakers

None scheduled at this time.

Seminars in academic year 2023

Date January 26, 2024   16:00–17:00
Speaker Yuya Kodama  (Tokyo Metropolitan University)
Title Divergence properties of Thompson-like groups
Place Dept. of Mathematics, Faculty of Science Bldg., F415  &  Zoom
Abstract Gersten defined a function called the divergence function for geodesic metric spaces. It is known that the order of this function is a quasi-isometric invariant (and hence is also a quasi-isometric invariant of finitely generated groups), and that the linearity of the order corresponds to the " degree of connectedness at infinity". In 2019, Golan--Sapir showed that the orders of functions of three groups, called the Thompson's groups, are linear. They then asked what is the order of the functions of Thompson-like groups. In this talk, I will discuss recent progress on this question.
Date January 19, 2024   16:00–17:00
Speaker Keisuke Himeno  (Hiroshima University)
Title Hyperbolic knots whose upsilon invariants are convex
Place Dept. of Mathematics, Faculty of Science Bldg., F415  &  Zoom
Abstract For a knot in $3$-sphere, the upsilon invariant is a concordance invariant which can be calculated from the full knot Floer complex. The invariant is a continuous piecewise linear function on $[0,2]$, and symmetric along the line $t=1$. Borodzik and Hedden gave the question of for which knots is the upsilon invariant a convex function. In this talk, we construct infinitely many mutually non-concordant hyperbolic knots whose upsilon invariants are convex.
Date December 15, 2023   16:00–17:00
Speaker Takuya Katayama  (Meiji University)
Title Bicorn curves on closed surfaces
Place Dept. of Mathematics, Faculty of Science Bldg., F415  &  Zoom
Abstract A bicorn curve between two simple closed curves on a closed surface is a concatenation of their subarcs whose endpoints coincide. In 2017, Przytycki and Sisto gave a simple proof for acylindrical hyperbolicity of the mapping class groups of closed orientable surfaces by using bicorn curves. In this talk, I will explain how to use bicorn curves in order to prove Hempel--Lickorish inequalities, Gromov hyperbolicity for the curve graphs and the bounded geodesic image theorem for nonorientable surfaces. The Hempel--Lickorish inequality states that the distance between two essential simple closed curves on a closed orientable surface is bounded above by a logarithmic function of the geometric intersection number. Masur and Minsky in 1999 described a quasi-geodesic word for any element of the mapping class groups. In their theory the role of Gromov hyperbolicity and the bounded geodesic image theorem is pivotal. This talk is based on joint work with Erika Kuno.
Date December 1, 2023   16:00–17:00
Speaker John R. Parker   (Durham University)
Title Margulis regions for screw-parabolic maps in real and complex hyperbolic space
Place Dept. of Mathematics, Faculty of Science Bldg., F415  &  Zoom
Abstract There is a celebrated result of Margulis about discrete subgroups of isometries of Riemannian manifolds with non-positive curvature. The result says that there is a constant only depending on the manifold so that, for each point on the manifold, the group generated by those isometries with displacement less than this constant is particularly simple. For many points this group is trivial, but the sets where it is infinite are called Margulis regions. In this talk, I will discuss the Margulis regions associated to screw parabolic maps with infinite order rotational part in hyperbolic 4-space. A particularly striking aspect of these results is the way they depend on the continued fraction expansion associated to the rotational part of the map, and the use of results from Diophantine approximation.
Date November 24, 2023   16:00–17:00
Speaker Kento Sakai   (Osaka University)
Title Degeneration of hyperbolic ideal polygons along harmonic map rays
Place Dept. of Mathematics, Faculty of Science Bldg., F415  &  Zoom
Abstract Let S be a surface of hyperbolic type and X be a complex structure on S. If S is a closed surface, Wolf constructed a homeomorphism between the Teichmüller space T(S) of S and the vector space of holomorphic quadratic differentials on X via harmonic maps between surfaces. A ray in the vector space of the holomorphic quadratic differentials determines the one-parameter family of hyperbolic surfaces through that homeomorphism. The one-parameter family is called by the harmonic map ray. Wolf showed that a harmonic map ray converges to an R-tree in the sense of equivariant Gromov-Hausdorff convergence if taking a lift to the universal covering and properly rescaling the hyperbolic metrics. In this talk, I will introduce the analogue result in the case that S is a hyperbolic ideal polygon, which is based on Gupta's result about the coordinate of the Teichmüller space of hyperbolic surfaces with crowns via harmonic maps. We consider a pointed Gromov-Hausdorff convergence instead of the equivariant version, since a hyperbolic ideal polygon does not have any nontrivial action by the surface group.
Date October 27, 2023   16:00–17:00
Speaker Jumpei Yasuda   (Osaka University)
Title BMW surfaces and Alexander theorem for surfaces in the 4-ball
Place Dept. of Mathematics, Faculty of Science Bldg., F415  &  Zoom
Abstract An n-knit is a tangle in a cylinder obtained from an n-braid by splicing for some crossings. BMW (Birman-Murakami-Wenzl) surfaces are properly embedded surfaces in the 4-ball which have motion pictures consisting of n-knits. These surfaces are generalizations of braided surfaces. In this talk, we will outline the construction of BMW surfaces from BMW charts due to Nakamura. As a main result, we state that every properly embedded surface in the 4-ball with a non-empty boundary is ambiently isotopic to a BMW surface.
Date September 8, 2023   14:00–15:00
Speaker Michal Jablonowski   (University of Gdansk)
Title Isotopic immersions of surfaces and Kirby moves
Place Dept. of Mathematics, Faculty of Science Bldg., F415  &  Zoom
Abstract We consider smooth immersions of surfaces in the four-space up to ambient isotopy, focusing on understanding their relationships with their complements in the ambient space. Beginning with a concise review of fundamental definitions and essential facts, we show how to obtain a Kirby diagram of the complement from a planar singular marked graph diagram of the immersed surface. We also discuss the consequences of this correspondence.
Date July 14, 2023   16:00–17:00
Speaker Naoki Sakata   (Ochanomizu University)
Title Searching entanglement parts of trefoil knots
Place Dept. of Mathematics, Faculty of Science Bldg., F415  &  Zoom
Abstract In ring polyethylene melts of trefoil knots, the structure is complex, with the crystalline and the amorphous states coexisting. In particular, as confirmed by united atom molecular dynamics simulations, the entanglements inhibit crystallization. In studying the detailed structures of the melts, it became necessary to investigate where the entanglement is located in a ring polyethylene polymer. In this talk, we will introduce the results of computer experiments to find the entangled parts of embedded trefoil knots. In addition, we will discuss how to give a mathematical definition of entanglement parts in general.
Date June 23, 2023   16:00–17:00
Speaker Han Yoshida   (National Institute of Technology (KOSEN), Gunma College)
Title Commensurators of Lobell polyhedra
Place Dept. of Mathematics, Faculty of Science Bldg., F415  &  Zoom
Abstract For a non-arithmetic Kleinian group $\Gamma < \mathrm{Isom}(\mathbb{H}^3)$, the commensurator $C(\Gamma) < \mathrm{Isom}(\mathbb{H}^3)$ is the (unique) maximal group commensurable with $\Gamma$. Few examples of computation of commensurators of cocompact Kleinian groups are known. Let $L_n$ be the L\"{o}bell polyhedron, which is a compact polyhedron with $2n+2$ faces, and $\Gamma(L_n)$ the subgroup of $\mathrm{Isom}(\mathbb{H}^3)$, generated by the reflections in the faces of $L_n$. In this talk, I will show the calculation of the commensurators $C(\Gamma(L_n))$ for sufficiently large $n$.
Date June 9, 2023   16:00–17:00
Speaker Katsunori Arai   (Osaka University)
Title On a groupoid rack coloring
Place Dept. of Mathematics, Faculty of Science Bldg., F415  &  Zoom
Abstract A spatial surface is a compact surface embedded in the 3-sphere. We assume that each connected component has non-empty boundary. Spatial surfaces are represented by diagrams of spatial trivalent graphs. In this talk, we introduce the notion of a groupoid rack, which is an algebraic structure that can be used for colorings of diagrams of oriented spatial surfaces. We show that for a given groupoid rack, the number of colorings is an invariant of oriented spatial surfaces. Furthermore, a groupoid rack has a universal property on colorings for diagrams of spatial surfaces.

Friday Seminar on Knot Theory in past academic years