Friday Seminar on Knot Theory(2022年度)

組織委員 阿蘇 愛理

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 12月16日 John Parker (Durham Univ.)
  1月27日 正井 秀俊 (東京工業大学)

日時 2023年1月27日(金)16:00~17:00
講演者(所属) 正井 秀俊 (東京工業大学)
タイトル Visualizing deformations of hyperbolic and complex structures on 4-punctured spheres.
場所 数学中講究室(理学部棟F415)& Zoom
アブストラクト We present movies and pictures of hyperbolic and complex structures on 4-punctured spheres.
The deformation space of both structures is known as Teichmueller space, and there are several natural paths that capture natural deformations of hyperbolic and complex structures. For example, Teichmueller geodesics and earthquake deformations will be discussed.
I will also talk about the motivations of those drawings in relation to hyperbolic volumes of fibered manifolds e.g. the fibered closure of braids of 3 strands.
日時 2022年12月16日(金)16:00~17:00
講演者(所属) John Parker (Durham Univ.)
タイトル Margulis regions for screw-parabolic maps
場所 数学中講究室(理学部棟F415)& Zoom
アブストラクト A famous result of Margulis says that there is a universal constant only depending on dimension with the following property. If G is a discrete group of hyperbolic isometries and x is a point then the elements of G that displace x by a distance less than the constant generate a nilpotent group. The thin part of the quotient orbifold is the collection of points where this nilpotent group is infinite. In this talk I will discuss the shape of the this part of a hyperbolic 4-manifold close associated to a screw-parabolic map with irrational rotational part. This involves results from Diophantine approximation in rather surprising ways.
日時 2022年12月2日(金)16:00~17:00
講演者(所属) 湯淺 亘 (OCAMI / RIMS)
タイトル State-clasp correspondence for skein algebras
場所 数学中講究室(理学部棟F415)& Zoom
アブストラクト For a compact oriented surface S with special points and intervals onthe boundary, we introduce the stated g-skein algebra and the claspedg-skein algebra of S for g=sp_4. Moreover, we show that the reducedversion of the stated g-skein algebra is isomorphic to theboundary-localization of the clasped g-skein algebra for a Lie algebrag=sl_2, sl_3 or sp_4. This isomorphism is a quantum counterpart of thetwo descriptions of the cluster algebra associated with the pair (g,S)in terms of the matrix coefficients of Wilson lines and cluster variables, respectively. This talk is based on joint work with TsukasaIshibashi (Tohoku Univ.).
日時 2022年11月25日(金)16:00~17:00
講演者(所属) 浅野 喜敬 (津山工業高等専門学校)
タイトル Some lower bounds for the Kirby-Thompson invariant of 4-manifolds
場所 数学中講究室(理学部棟F415)& Zoom
アブストラクト A trisection is a decomposition of a closed 4-manifold X into a 3-tuple of 4-dimensional 1-handlebodies, which was introduced by Gay and Kirby. Kirby and Thompson defined an invariant of X by using trisections. This invariant is called the Kirby-Thompson invariant. In this talk, we give some lower bounds for the Kirby-Thompson invariant of certain 4-manifolds. As an application, we find the first example of a 4-manifold whose Kirby-Thompson invariant is non-zero. This is a joint work with Hironobu Naoe (Chuo University) and Masaki Ogawa (Saitama University).
日時 2022年11月18日(金)16:00~17:00
講演者(所属) 久野 恵理香 (大阪大学)
タイトル Quasi-isometric embeddings induced by the orientation double coverings
場所 数学中講究室(理学部棟F415)& Zoom
アブストラクト Birman--Chillingworth firstly proved that each mapping class group of a nonorientable surface is a subgroup of the mapping class group of the doule covering orientable surface. We prove that this natural injective homomorphism is a quasi-isometric embedding by using semihyperbolicity of the (extended) mapping class groups of the orientable surfaces. This is a joint work with Takuya Katayama.
日時 2022年10月28日(金)16:00~17:00
講演者(所属) 長谷川 耀 (大阪大学)
タイトル Gromov boundaries of non-proper hyperbolic geodesic spaces
アブストラクト In a proper hyperbolic geodesic space, it is well known that the sequential boundary can be identified as topological spaces with the geodesic boundary. We show that in a (not necessarily proper) hyperbolic geodesic space, the sequential boundary can be identified as topological spaces with the quasi-geodesic boundary.
日時 2022年7月22日(金)16:00~17:00
講演者(所属) 軽尾 浩晃 (学習院大学)
タイトル Quantum trace maps for LRY skein algebras
アブストラクト Based on Bonahon and Wong's works, Le formulated stated skein algebras to understand quantum trace maps. On the other hand, Roger--Yang introduced Roger--Yang skein algebras to give an explicit connection between skien algebras and decorated Teichmuller spaces. As a generalization of stated skein algebras and RY skein algebras, one can consider Le--Roger--Yang skein algebras. In this talk, we show the key idea to construct the quantum trace maps for LRY skein algebras and what the LRY skein algebra of an elementary surface is. This is a joint work with W. Bloomquist and Thang T. Q. Le at Georgia Tech.
日時 2022年7月8日(金)16:00~17:00
講演者(所属) 谷口 雄大 (大阪大学)
タイトル The knot quandle of the $n$-twist spun knot is a central extension of the knot $n$-quandle.
アブストラクト Given an oriented $m$-knot $K$, we have the knot quandle of $K$, which is an analogy of the knot group of $K$. Inoue showed that the knot quandle of $n$-twist spun trefoil is a central extension of the Schl\"{a}fli quandle related to $\{ 3,n\}$. In this talk, we generalize the Inoue's result. More precisely, we show that for any $1$-knot $K$, the knot quandle of the $n$-twist spun $K$ is a central extension of the knot $n$-quandle of $K$. This is a joint work with Kokoro Tanaka (Tokyo Gakugei University).
日時 2022年7月1日(金)16:00~17:00
講演者(所属) 安田 順平 (大阪大学)
タイトル Computation of the knot symmetric quandle and the plat index for surface-links
アブストラクト A plat form for links is a presentation of a classical link using a braid. We can apply this presentation to surface-links, using a braided surface instead of a braid, and prove that every surface-link has a plat form presentation. In this talk, we show how to compute the knot symmetric quandle of a surface-link using a plat form.
As an application, for any positive integer m, we give infinitely many surface-knots with the plat index m.
日時 2022年6月17日(金)16:00~17:00
講演者(所属) 小川 将輝 (埼玉大学)
タイトル On the reducibility of handlebody decompositions.
アブストラクト Haken showed that any Heegaard splittings of reducible 3-manifolds are reducible. This is well-known and called Haken’s lemma.
We can consider a decomposition of a 3-manifold with three handlebodies. We call a such decomposition a handlebody decompositions. This is a generalization of a Heegaard splitting. In this talk, we introduce an operation called a connect sum of a handlebody decomposition. After that, we consider the handlebody decompositions of the connected sum of two lens spaces and show that some of them are obtained by connect summing two handlebody decompositions.
日時 2022年6月3日(金)16:00~17:00
講演者(所属) 岡崎 真也 (奈良教育大学)
タイトル On the crossing number of constituent links of a handlebody-knot
アブストラクト A handlebody-knot is a handlebody embedded in the 3-sphere.
When the genus 2 handlebody-knot is cut open with a separating disk, a two component knotted solid tori appears. This is regarded as a 2-component link and is called a constituent link of the handlebody-knot.
In this talk, we introduce a generalization of the degree of Laurent polynomial. We show that the generalized degree of the Alexander polynomial of a pair consisting of a genus 2 handlebody-knot and its meridian system give a lower bound of the crossing number of constituent links of a genus 2 handlebody-knot.
日時 2022年5月13日(金)16:00~17:00
講演者(所属) 佐々木 東容 (学習院大学)
タイトル Currents on cusped hyperbolic surfaces and denseness property
アブストラクト The space GC(S) of geodesic currents on a hyperbolic surface S can be considered as a completion of the set of weighted closed geodesics on S when S is compact, since the set of rational geodesic currents on S, which correspond to weighted closed geodesics, is a dense subset of GC(S). We proved that even when S is a cusped hyperbolic surface with finite area, GC(S) has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on S but also to weighted geodesics connecting two cusps. In addition, we proved that the space of subset currents on a cusped hyperbolic surface, which is a generalization of geodesic currents, also has the denseness property of rational subset currents.
In this talk, we will talk about the characterization of rational geodesic currents and subset currents by using basic hyperbolic geometry.
最終更新日: 2022年12月15日