Perturbative expansion of quantum invariants

3-流形的量子不变量定义在紧李群上，最早由维滕预言，并由列什季欣和图拉耶夫严格建立。虽然不变量的定义是复值的，但它被认为是代数整数。Murakami在李群SU（2）中证实了这一点，Ohtsuki从其中通过所谓的“代数扰动”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它李群构造这种3-流形的微扰不变量，但这一课题是由Le独立完成的。因此，我们继续讨论了这些不变量的几何应用，得到了以下结果：1.三维流形中曲面不可压缩性的判定尽管不可压缩曲面在三维流形理论中占有重要地位，但如何判定流形中曲面的不可压缩性一直是一个难题。本文利用量子不变量导出的映射类群的表示矩阵，建立了流形中非分离曲面不可压缩性的一些判据。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分裂不变量在这种等价下的行为是平凡的。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θ-曲线的多项式不变量引入了三维空间中θ-曲线的新的多项式不变量。这些不变量是绝对可计算的，并且可以检测立体化学家感兴趣的图的手性。

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.