Title: Quantum Invariants, Skein Modules, and Periodicity of 3-Manifolds (Ph.D. Dissertation)

Author: Maxim V. Sokolov
Dissertation Director: Jozef H. Przytycki

Preprint: GWUM-2000-2 

Abstract:

We study various quantum invariants and skein modules of 3-manifolds and their relationship to each other and to classical invariants of 3-manifolds. We study, in particular, how quantum invariants and classical invariants (homology) reflect periodicity of 3-manifolds. The first chapter is devoted to Turaev-Viro invariants. We show that each Turaev-Viro invariant is a sum of three invariants, study properties of the summand invariants and establish certain relationships. We solve a conjecture due to Kauffman and Lins and provide a simple and effective criterion answering whether two given lens spaces are distinguished by Turaev-Viro invariants or not. Chapter 2 is probably the most interesting chapter of the theses. We develop a special surgery presentation for p-periodic 3-manifolds and prove the following result: if a closed orientable 3-manifold M admits an action of a cyclic group Z_p where p is an odd prime integer and the fixed point set of the action is S^1 then H_1(M; Z_p) is not equal to Z_p. We also provide a similar criterion for the 2-periodic rational homology 3-spheres. Chapter 3 studies how quantum invariants reflect periodicity of 3-manifolds. We consider Dijkgraaf-Witten invariants and the simplest known quantum invariants discovered by H. Murakami, T. Ohtsuki, and M. Okada. In the last chapter, Chapter 4, we study some simple skein modules (S and S_2) and their relationship with quantum invariants. We show how to construct Murakami-Ohtsuki-Okada invariants from these skein modules using Lickorish's method. In Section 3 of Chapter 4 we will show that if q is a root of unity then no other invariants can be obtained using Lickorish method on S and S_2. We also study Deloup invariants and their relationship with the tensor product of two copies of S_2(M;R, q). In Appendix A we show how to define Dijkgraaf-Witten invariants using singular triangulations and simple 2-skeletons of 3-manifolds. In Appendix B we present tables with values of Turaev-Viro invariants for 239 closed oriented 3-manifolds of small complexity. The values are presented as cyclotomic integers. In Appendix C we show how to determine linking pairing of a rational homology 3-sphere from its rational surgery presentation.