Invariants of knots and 3-manifolds

组织研究结点和3流形的不变量。我从把结的2环多项式看作结的补的无限循环覆盖的“等变Casson不变量”的观点出发，利用结的Seifert曲面的一个棘的* 4次有限型不变量，给出了结的2环多项式。进一步研究了结点补无穷循环覆盖的等变量子不变量，统一了沿结点分支的有限覆盖的量子不变量。进一步，从研究结的补无穷循环覆盖的等变量子U(1)不变量的观点出发，从结的手术演示的等变连接矩阵构造了结的不变量。此外，在一定情况下，我提出了具有第一个Betti数1的3流形的微扰不变量，作为3流形量子不变量的算术微扰展开。我与本研究的合作研究者Kazuo Habiro共同组织了一次低维拓扑研讨会。主讲人是Yoko Mizuma， Takahito Kuriya, Soeren Hansen, Alexander Stoimenow, Vladimir Turaev, Alexis Virelizier, Gwenael Massuyeau, Masamichi Takase，他们的演讲是关于结和3流形不变量领域的高级主题。我认为他们的演讲非常好，无论是从他们与我和合作研究者的共同研究角度，还是从他们与年轻研究者如研究生的研究互动角度。

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.