Perturbative expansion of quantum invariants
Project/Area Number  09640118 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Geometry

Research Institution  KYUSHU UNIVERSITY 
Principal Investigator 
YOKOTA Yoshiyuki Kyushu Univ., Graduate School of Mathematics, Assistant Professor, 大学院・数理学研究科, 講師 (40240197)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,700,000 (Direct Cost : ¥2,700,000)
Fiscal Year 1998 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1997 : ¥2,000,000 (Direct Cost : ¥2,000,000)

Keywords  quantum invariant / incompressible surface / Heegaard splitting / rheta_mcurve / 量子不変量 / 非圧縮曲面 / Heegaard分解 / θm曲線 / Heegaar分解 / 三次元多様体 
Research Abstract 
The quantum invariants of 3manifolds, which are defined for compact Lie groups, was first predicted by Witten, and rigorously established by Reshtikhin and Turaev. Although the invariants are complexvalued by definition, it has been believed to be algebraic integers. This is confirmed for Lie group SU(2) by Murakami, _from which Ohtsuki extracted an infinite series of 3manifold invariants by socalled "algebraic perturbation". The purpose of this research was to construct such perturbative invariants of 3manifolds for the other Lie groups, but this subject is independently acheived by Le. Thus, I proceed to the geometric application of the invariants, and obtain the following results. 1.Criteria for the incompressibility of surfaces in 3manifolds Although incompressible surfaces play important roles in 3manifold theory, it has been difficult to detect the incom pressibility of surfaces in manifolds. In this research, some criteria for such incompressibility of nonseparating surfaces in manifolds are established in terms of representation matrices of mapping class groups derived from quantum invariants. 2.New invariants for Heegaard splittings of 3manifolds It is wellknown that two Heegaard splittings of a 3manifold are stably equivalent, but known invariants of Heegaard splittings behaves trivially under such equivalence. In this research some invariants are defined in terms of the elementary divisors of representation matrices of mapping class groups which behave nontrivially under stably equivalence. 3.Polynomial invariants for thetamcurves in 3space New polynomial invariants for thetamcurves in 3space are introduced. The invariants are definitely computable, and can detect the chirality of graphs in which stereochemists are interested. これは副産物というべき研究ですが、ユニタリー群に対応する量子不変量を定式化するときに用いた議論を利用して、θ_m曲線と呼ばれるグラフの、空間への埋め込みに対する多項式不変量を構成しました。この不変量は、定義の簡単さもさることながら、グラフのキラル性を判定できる点が特徴で、分子の位相的立体異性体の研究など、今後様々な応用が期待されます。

Report
(4results)
Research Output
(12results)