Perturbative expansion of quantum invariants

量子不变量的微扰展开

基本信息

批准号：
09640118
负责人：
YOKOTA Yoshiyuki
金额：
$ 1.73万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640118/
关键词：
quantum invariant incompressible surface Heegaard splitting rheta_m-curve Heegaar分解三次元多様体

项目摘要

The quantum invariants of 3-manifolds, which are defined for compact Lie groups, was first predicted by Witten, and rigorously established by Reshtikhin and Turaev. Although the invariants are complex-valued 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 3-manifold invariants by so-called "algebraic perturbation". The purpose of this research was to construct such perturbative invariants of 3-manifolds 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 3-manifoldsAlthough incompressible surfaces play important roles in 3-manifold theory, it has been difficult to detect the incom- pressibility of surfaces in manifolds. In this research, some criteria for such incompressibility of non-separating 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 3-manifoldsIt is well-known that two Heegaard splittings of a 3-manifold 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 thetam-curves in 3-spaceNew polynomial invariants for thetam-curves in 3-space are introduced. The invariants are definitely computable, and can detect the chirality of graphs in which stereo-chemists are interested.

为紧李群定义的 3 流形量子不变量首先由 Witten 预测，并由 Reshtikhin 和 Turaev 严格建立。尽管根据定义，不变量是复值，但它被认为是代数整数。 Murakami 对李群 SU(2) 证实了这一点，Ohtsuki 通过所谓的“代数微扰”从中提取了无限级数的 3 流形不变量。这项研究的目的是为其他李群构造这样的3-流形的微扰不变量，但这个课题是由Le独立实现的。因此，我继续对不变量进行几何应用，并得到以下结果。 1. 3-流形中表面不可压缩性的判据虽然不可压缩表面在3-流形理论中起着重要作用，但很难检测到不可压缩表面。流形中表面的可压性。在这项研究中，流形中非分离表面的不可压缩性的一些准则是根据从量子不变量导出的映射类群的表示矩阵来建立的。2.3-流形Heegaard分裂的新不变量众所周知，两个Heegaard 3 流形的分裂是稳定等价的，但 Heegaard 分裂的已知不变量在这种等价性下表现得很平常。在这项研究中，一些不变量是根据映射类群的表示矩阵的初等除数来定义的，这些映射类群在稳定等价下表现得非平凡。 3. 3 空间中 thetam 曲线的多项式不变量 3 空间中 thetam 曲线的新多项式不变量是介绍了。这些不变量绝对是可计算的，并且可以检测立体化学家感兴趣的图的手性。