Invariants of knots and 3-manifolds

组织研究结点和3流形的不变量。我编辑了2001年9月组织的“节和3流形的不变量”研讨会的会议记录，并在《几何和拓扑专论》杂志上发表了会议记录。特别是，我编辑了一份问题清单，这是根据研讨会问题会议中给出的问题制作的，并将其作为会议记录的一部分发表。我研究了Kontsevich不变量的循环展开，特别是2环多项式，它呈现了循环展开的2环部分。我从将结的2环多项式视为结补的无限循环覆盖的等变Casson不变量的观点出发，以结的Seifert曲面的棘的有限型不变量的形式给出了结的2环多项式。通过具体的计算，我得到了一个结的二环多项式的度的界，这个界是结的两倍的格，更多地介绍高斯图。进一步，我给出了环面结和1属结的2环多项式的明确表示。此外，我展示了一个2环多项式的布线公式，它给出了给定结的任何电缆结的2环多项式。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Sergei Duzhin， Kazuhiro Hikami, Andrew Kricker, Julien Marche, Jean-Baptiste Meilhan, Gregor Masbaum, Jorgen Andersen, Yoshiyuki Yokota, Jozef Przytycki，他们的演讲是关于节和3流形不变量领域的高级主题。特别是，在这项研究的资金支持下，Kricker和Marche在RIMS呆了两个星期。我觉得他们的谈话和停留都很好，无论是从他们和我以及合作研究者的共同研究来看，还是从他们和年轻研究者比如研究生的研究互动来看。少

I organized researches on invariants of knots and 3-manifolds.I edited the proceedings of the workshop "Invariants of Knots and 3-Manifolds", which I organized in September 2001, and published the proceedings from the journal Geometry and Topology Monographs. In particular, I edited a list of problems, which was made based on problems given in the problem sessions of the workshop, and published it as a part of the proceedings.I studied the loop expansion of the Kontsevich invariant, in particular, the 2-loop polynomial, which presents the 2-loop part of the loop expansion. I presented the 2-loop polynomial of a knot in terms of finite type invariants 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 knot complement. I obtained a bound of the degree of the 2-loop polynomial of a knot by twice the genus of the knot, by calculating the presentation concretely … More introducing Gaussian diagrams. Further, I gave explicit presentations of the 2-loop polynomial for the torus knots and the knots of genus 1. Furthermore, I showed a cabling formula for the 2-loop polynomial, which gives the 2-loop polynomial for any cable knot of a given knot.I organized a low-dimensional topology seminar, jointly with Kazuo Habiro, who is a co-investigator of this research. The speakers were Sergei Duzhin, Kazuhiro Hikami, Andrew Kricker, Julien Marche, Jean-Baptiste Meilhan, Gregor Masbaum, Jorgen Andersen, Yoshiyuki Yokota, Jozef Przytycki, and their talks were on advanced topics in the area of invariants of knots and 3-manifolds. In particular, by financial supports from the grant of this research, Kricker and Marche stayed at the RIMS in two weeks. I think their talks and stays 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. Less