# Friday Seminar on Knot Theory

#### Friday Seminar on Knot Theory (2021)

Organizers: Shin'ya Okazaki・Airi Aso

List of Friday Seminar on Knot Theory by Year

Speaker | Keegan Boyle (The University of British Columbia) |
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Title | An equivariant signature function for periodic and strongly invertible knots |

Date | January 28 (Fri.) 15：30～16：30 |

Abstract | The knot signature is an important classical invariant of knots which gives a lower bound on the smallest genus of orientable surface a knot can bound in the 4-ball. In this talk I will discuss recent work, some with Ahmad Issa and some with Antonio Alfieri, in which we define an analogous invariant, the equivariant signature, and show that it gives a lower bound on the smallest genus of an invariant surface which a symmetric knot can bound in the 4-ball, for an appropriately restricted class of surfaces. The key technical tool we use is the Atiyah-Singer signature theorem applied to 4-manifolds. |

Speaker | Naoki Sakata (Saitama University) |
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Title | Stabilizations on handlebody decompositions and polycontinuous patterns |

Date | January 14 (Fri.) 16：00～17：00 |

Abstract | Koenig introduced the concept of trisections of closed orientable 3-manifolds, which is an embedded branched surface dividing the manifold into three handlebodies. He also defined an operation,called stabilization, on trisections and proved an analogue of Reidemeister-Singer’s theorem for trisections. This talk will introduce “handlebody decompositions,” which is a decomposition of a closed orientable 3-manifold into multiple handlebodies. We will also define stabilizations and prove an analogue of Reidemeister-Singer’s theorem for handlebody decompositions. The motivation for this work comes from the study of materials science. It is known that a Heegaard surface of the 3-dimensional torus corresponds to an interface mediated in a particular type of self-assembled diblock copolymers, called a bicontinuous pattern. In this talk, we will generalize this concept to define polycontinuous patterns and discuss an analogue of Reidemeister-Singer’s theorem for them. |

Speaker | Daiki Iguchi (Hiroshima University) |
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Title | Distance and the Goeritz groups of bridge decompositions |

Date | December 17 (Fri.) 16：00～17：00 |

Abstract | A bridge decomposition of a link in a closed orientable 3-manifold is a decomposition of the link into two "trivial tangles" across a Heegaard surface of the 3-manifold. The Goeritz group of a bridge decomposition is defined to be the group of isotopy classes of orientation-preserving diffeomorphisms of the 3-manifold that preserve the decomposition. In this talk, we show that if the distance of a bridge decomposition is at least 6, then its Goeritz group is a finite group. This is a joint work with Yuya Koda (Hiroshima University). |

Speaker | Akihiro Takano (The University of Tokyo) |
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Title | The Long-Moody construction and twisted Alexander invariants |

Date | December 10 (Fri.) 16：00～17：00 |

Abstract | The Long-Moody construction is a method of constructing a new representation of the braid group from a representation of the semidirect product of the braid group and the free group. In this talk, we show that its matrix presentation is described by the Fox derivation, and also a relation with twisted Alexander invariants. |

Speaker | Naoki Kimura (Waseda University) |
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Title | Legendrian knots and rack coloring invariants |

Date | December 3 (Fri.) 16：00～17：00 |

Abstract | Kulkarni-Prathamesh (2017) introduced a new invariant of Legendrian knots by using rack colorings.Ceniceros-Elhamdadi-Nelson (2021) defined a Legendrian rack and generalized the invariant. In this talk, we consider a further generalization of the invariant, which we call a bi-Legendrian rack coloring.We show that bi-Legendrian rack coloring numbers can distinguish all Legendrian unknots with the same Thurston-Bennequin number. We also consider pairs of Legendrian knots which cannot be distinguished by bi-Legendrian rack coloring numbers. |

Speaker | Takuya Katayama (Gakushuin University) |
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Title | Pure braid groups in mapping class groups of surfaces |

Date | November 19 (Fri.) 16：00～17：00 |

Abstract | In this talk, we give a necessary and sufficient condition for embedding pure braid groups into the mapping class groups of surfaces. Our methods for embedding pure braid groups into the mapping class groups of surfaces are based on the Birman--Hilden theory and Paris--Rolfsen's work. On the other hand, right-angled Artin groups in mapping class groups tell us size-like features of the mapping class groups---the mapping class groups can be distinguished from each other by their right-angled Artin spectra. Using such spectra, we prove the desired non-existence of embeddings. If time allows, we also discuss virtual injections from the mapping class groups of tori. |

Speaker | Tomo Murao (Waseda University) |
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Title | On constituent links of genus 2 handlebody-knots |

Date | November 5 (Fri.) 16：00～17：00 |

Abstract | A handlebody-knot is a handlebody embedded in the 3-sphere. A constituent link of a genus 2 handlebody-knot H is the spine of two solid tori obtained from H by removing an open regular neighborhood of a separating essential disk in H. In general, a genus 2 handlebody-knot has infinite constituent links. In this talk, we provide methods to detect constituent links of a genus 2 handlebody-knot in terms of coloring theory. |

Speaker | Abdoul Karim SANE (Université Cheikh Anta Diop) |
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Title | Surgery graphs on unicellular maps |

Date | November 5 (Fri.) 17：15～18：15 |

Abstract | There are well-known complexes associates to surfaces: curves complexes, arcs complexes, graph of pants decomposition etc... and their studies are related to interesting question on low dimensional topology. In this talk we will introduce a family of new graphs called surgery graphs on unicellular maps. We will discuss some interactions with the mapping class group. |

Speaker | Katsumi Ishikawa (Kyoto University) |
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Title | A spectral sequence on quandle homology |

Date | October 8 (Fri.) 16：00～17：00 |

Abstract | Quandle homology theory was introduced as a quotient of rack homology and its low-dimensional part especially plays an important role in knot theory. In this talk, we show that for any quandle a covering of the quandle space is homotopy equivalent to (a refinement of) the extended quandle space, which implies that it has the shifted quandle homology groups. Then, under a certain condition satisfied for any generalized Alexander quandles, we can consider the Cartan-Leray spectral sequence of the covering to compute homology groups. In fact, we determine the all homology groups of any connected dihedral quandles and the third homology of Alexander quandles on finite fields. |

Speaker | Takuya Ukida (National Institute of Technology, Anan College) |
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Title | Infinite number of genus zero Lefschetz fibrations on the Akbulut-Yasui plugs |

Date | July 16 (Fri.) 16：00～17：00 |

Abstract | We construct a genus zero PALF structure on each of plugs introduced by Akbulut and Yasui and describe the monodromy as a positive factorization in the mapping class group of a fiber. |

Speaker | Kodai Wada (Kobe University) |
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Title | Forbidden detour moves for virtual links |

Date | June 25 (Fri.) 16：00～17：00 |

Abstract | In virtual knot theory, there are two disallowed moves, known as the forbidden moves. In the study of forbidden moves, Kanenobu and Nelson independently introduced a local move, called a forbidden detour move. This move is realized by the forbidden moves. In particular, the equivalence relation on virtual links generated by forbidden detour moves implies the one by forbidden moves. This gives rise to the question: is the converse true? Recently, Yoshiike and Ichihara proved that the question is true for virtual knots. In this talk, we show that it is true for 2-component virtual links by classifying them up to forbidden detour moves. |

Speaker | Nobutaka Asano (Tohoku University) |
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Title | 4-manifolds admitting simplified genus-2 trisections with prescribed vertical 3-manifolds |

Date | June 4 (Fri.) 16：00～17：00 |

Abstract | A trisection of Gay-Kirby is a decomposition of a closed 4-manifold into three 4-dimesional 1-handlebodies. They proved the existence of a trisection for any closed 4-manifold by constructing a stable map from the 4-manifold to the real plane, called a trisection map. We focus on the 3-manifolds obtained as the preimages of arcs on the real plane for simplified genus-2 trisection maps, called vertical 3-manifolds. Any vertical 3-manifold is given as a connected sum of finite copies of six basic vertical 3-manifolds and S2×S1. We show that non-trivial 6-tuples of vertical 3-manifolds determine the source 4-manifolds uniquely up to orientation reversing diffeomorphisms. If time permits, the speaker's recent research will be presented. |

Speaker | Tamotsu Basseda (Osaka City University) |
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Title | A normalization of A_2 bracket invariant for spatial links |

Date | May 14 (Fri.) 16：00～17：00 |

Abstract | A_2 bracket invariant defined by Kuperberg can be thought as a regular isotopy invariant for spatial graphs. I made it ambient isotopy invariant. I show how it can be normalized and that the ambient isotopy invariant has some natural properties. |

Speaker | Masaki Ogawa(Saitama University) |
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Title | Stably equivalence of multibranched handlebody decomposition |

Date | May 7 (Fri.) 16：00～17：00 |

Abstract | A decomposition of a 3-manifold with some handlebodies and some properties is called a handlebody decomposition. We studied the case where the union of the intersection of handlebodies is a simple polyhedron. In this talk, we introduce the new class of handlebody decomposition called a mulitibranched handlebody decomposition. Reidemeister-Singer theorem says two Heegaard splittings can be isotopic after finite number of stabilizations. Koenig introduced a concept of a decomposition of 3-manifolds with three handlebodies called a trisection of a 3-manifold and he showed the stably equivalence of a trisection of a 3-manifold. After that, we generalized it into a decomposition with n handlebodies. We showed that a handlebody decomposition whose partition is a simple polyhedron is stably equivalent by stabilizations and some moves with keeping the partition as a simple polyhedron. In this talk, we shall consider the stably equivalence problem of multibranched handlebody decomposition and show stably equivalence if the number of handlebodies is 4. |

Speaker | Hiroaki Karuo (Kyoto University) |
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Title | Degenerations of Muller--Roger--Yang skein algebras |

Date | April 30 (Fri.) 16：00～17：00 |

Abstract | For oriented surfaces with ideal triangulations, Bonahon and Wong gave embeddings of their ordinary skein algebras into quantum tori, called quantum trace maps. After that, Le and Yu gave quantum trace maps for stated skein algebras, a generalization of ordinary skein algebras. These embeddings let us know several properties of (stated) skein algebras. In this talk, we focus on a generalization of Muller skein algebras and Roger--Yang skein algebras and give embeddings of "degenerations'' of them into quantum tori. This is joint work with Thang Le (Georgia Institute of Technology). |

Speaker | Yuta Taniguchi (Osaka University) |
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Title | f-twisted Alexander matrices of fundamental quandles |

Date | April 23 (Fri.) 16：00～17：00 |

Abstract | A quandle is an algebraic structure defined on a set with a binary operation whose axioms correspond to Reidemeister moves. D. Joyce and S. V. Matveev associated a quandle to a link, which is called the fundamental quandle. Since then many link invariants using fundamental quandles have been introduced and studied. Recently, A. Ishii and K. Oshiro introduced the f-twisted Alexander matrix, which is a quandle version of the twisted Alexander matrix. They showed that the twisted Alexander matrix can be recovered from the f-twisted Alexander matrix. In this talk, we study a relationship between f-twisted Alexander matrices and quandle cocycle invariants. As an application, we show that a square knot and a granny knot, which have the same knot group, are distinguished by an f-twisted Alexander matrix. |

Speaker | Hideo Takioka (Kyoto University) |
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Title | On the first coefficient Kauffman polynomial of a knot |

Date | April 16 (Fri.) 16：00～17：00 |

Abstract | In this talk, we recall a skein relation of the first coefficient Kauffman polynomial for knots. By using the skein relation, we show that there exist infinitely many knots whose Whitehead doubles have the trivial first coefficient Kauffman polynomial. |