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。我编辑了2001年9月我组织的研讨会《纽结和3-流形的不变量》的会议记录，并发表了来自《几何和拓扑专著》的会议记录。特别是，我编辑了一个问题清单，这是根据研讨会的问题会议中给出的问题制作的，并将其作为过程的一部分发布。我研究了康采维奇不变量的回路展开，特别是表示回路展开的2回路部分的2回路多项式。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曲面的脊线的有限型不变量，给出了纽结的2-圈多项式.通过具体计算纽结的表示…，得到了纽结的2-圈多项式的次数是纽结亏格的两倍的界更多介绍高斯图。此外，我还给出了环结和亏格1的结点的2-圈多项式的显式表示。此外，我还给出了2-圈多项式的布线公式，它给出了给定纽结的任何缆结的2-圈多项式。我与本研究的共同研究员Kazuo Habiro共同组织了一个低维拓扑研讨会。演讲者是Sergei Duzhin、Kazuhiro Hikami、Andrew Kricker、Julien Marche、Jean-Baptiste Meilhan、Gregor Masbaum、Jorgen Andersen、Yoshiyuki Yokota、Jozef Przytycki，他们的演讲是关于纽结不变量和3-流形领域的高级主题。特别值得一提的是，在这项研究的资助下，克里克和马尔奇在篮筐呆了两周。我认为他们的谈话和停留非常好，从他们与我和合作研究员的联合研究的角度，从他们与研究生等年轻研究人员的研究互动的角度来看，我认为他们的谈话和停留非常好。较少

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