| Research Abstract |
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
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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
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