# Friday Seminar on Knot Theory

## Friday Seminar on Knot Theory

Online participation is possible. Please send the online registration form from the following link:

Registration for Friday Seminar on Knot Theory via Zoom

Organizer: Hirotaka Akiyoshi (akiyoshi@omu.ac.jp)

## Upcoming speakers

None scheduled at this time.

## Seminars in academic year 2023

Date | January 26, 2024 16:00–17:00 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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. |