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月组织的“纽结和3-流形的不变量”研讨会的会议记录，并在《几何和拓扑学专著》杂志上发表了会议记录。特别是，我编辑了一个问题列表，这是根据在讲习班的问题会议上给出的问题，并将其作为会议录的一部分发表。我研究了Kontsevich不变量的循环扩展，特别是2-loop多项式，它表示循环扩展的2-loop部分。我提出的2-loop多项式的节的塞弗特表面的脊椎的有限型不变量的节，从我把2-loop多项式的纽结的无限循环覆盖的纽结补的等变卡森不变量的观点。通过具体的计算，得到了纽结的2-圈多项式的次数以纽结亏格的2倍为界的一个界 关于我们 介绍高斯图。此外，我给出了明确介绍的2-循环多项式的环面结和结的亏格1。此外，我还展示了2-loop多项式的布线公式，它给出了给定结的任何电缆结的2-loop多项式。我与本研究的合作研究者Kazuo Habiro共同组织了一个低维拓扑研讨会。发言者谢尔盖Duzhin，Kazuhiro Hikami，安德鲁Kricker，朱利安马尔凯，让巴蒂斯特梅尔汉，格雷戈尔Masbaum，约根安徒生，Yoshiyuki横田，约瑟夫Przytycki，他们的会谈是先进的主题领域的不变量的结和3流形。特别是，通过这项研究的资助，Kricker和马尔凯在RIMS呆了两周。我认为，从他们与我和共同研究者之间的共同研究的角度来看，以及从他们与研究生等年轻研究人员之间的研究互动的角度来看，他们的谈话和停留都非常好。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