Invariants of knots and 3-manifolds

我组织了关于纽结和3-流形的不变量的研究。从把纽结的2-圈多项式视为纽结补的无限循环覆盖的“等变Casson不变量”的观点出发，利用纽结的Seifert曲面的脊线的*4次有限类型不变量，给出了纽结的2-圈多项式.进一步，研究了纽结补的无限循环覆盖的等变量子不变量，统一了沿纽结分支的有限覆盖的量子不变量。进一步地，从纽结的运算表示的等变连接矩阵出发，从研究纽结补的无限循环覆盖的等变量子U(1)不变量的角度，构造了纽结的不变量。在此基础上，将三维流形的第一个Betti数为1的微扰不变量表示为三维流形量子不变量的算术微扰展开式，并与本研究的研究员Kazuo Habiro共同组织了一次低维拓扑研讨会。演讲者是水间洋子、高屋、苏伦·汉森、亚历山大·斯托伊门诺、弗拉基米尔·图拉耶夫、亚历克西斯·维雷利齐耶、格温内尔·Massuyeau、Masamichi Takase，他们的演讲是关于纽结和3-流形不变量领域的高级主题。我认为他们的谈话非常好，从他们与我和协作者共同研究的角度，从他们与研究生等年轻研究人员的研究互动的角度来看，他们的谈话非常好。

I organized researches on invariants of knots and 3-manifolds. I presented the 2-loop polynomial of a knot by using finite type invariants of degree * 4 of a spine of a Seifert surface of the knot, from the viewpoint that I regard the 2-loop polynomial of a knot as an "equivariant Casson invariant" of the infinite cyclic cover of the complement of the knot. Further, I studied equivariant quantum invariants of the infinite cyclic cover of the complement of a knot, which unify quantum invariants of finite covers branched along the knot. Further, I constructed invariants of knots from equivariant linking matrices of their surgery presentations, from the viewpoint of studying equivariant quantum U(1) invariant of the infinite cyclic covers of the complements of the knots. Furthermore, I formulated a perturbative invariant of a 3-manifold with the first Betti number 1 in a certain case, as the arithmetic perturbative expansion of quantum invariants of the 3-manifold.I organized a low-dimensional topology seminar, jointly with Kazuo Habiro, who is a co-investigator of this research. The speakers were Yoko Mizuma, Takahito Kuriya, Soeren Hansen, Alexander Stoimenow, Vladimir Turaev, Alexis Virelizier, Gwenael Massuyeau, Masamichi Takase, and their talks were on advanced topics in the area of invariants of knots and 3-manifolds. I think their talks were very good, from the viewpoint of joint researches between them and me and the co-investigator, and from the viewpoint of research interactions between them and young researchers such as graduate students.