Perturbative expansion of quantum invariants

三维流形的量子不变量是由Witten首先预言的，并由Reshtikhin和Turaev严格建立的。尽管不变量从定义上来说是复值的，但它一直被认为是代数整数。村上对Lie群SU(2)证实了这一点，大树通过所谓的“代数扰动”从SU(2)中提取了一个3-流形不变量的无穷级数。本研究的目的是为其他李群构造3-流形的扰动不变量，但这一课题是由Le独立完成的。1.三维流形中曲面的不可压缩判据虽然不可压缩曲面在三维流形理论中扮演着重要的角色，但在流形中却很难检测到曲面的不可压缩性。利用由量子不变量导出的映射类群的表示矩阵，建立了流形上非分离曲面不可压缩的一些判据。2.三维流形的Heegaard分裂的新不变量众所周知，三维流形的两个Heegaard分裂是稳定等价的，但已知的Heegaard分裂的不变量在这样的等价下表现得很平凡。本文利用映射类群的表示矩阵的初等因子定义了一些在稳定等价下表现非平凡的不变量。3.三维空间中曲线的多项式不变量引入了三维空间中曲线的新的多项式不变量。这些不变量绝对是可计算的，可以检测立体化学家感兴趣的图形的手性。

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.