#### Title

#### Date of Award

2000

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Patrick Gilmer

#### Abstract

Let k be a subring of the field of rational functions in x, v, s which contains x+/-1, v+/-1, s+/-1. If M is an oriented 3-manifold, let S(M) denote the Homflypt skein module of M over k. This is the free k-module generated by isotopy classes of framed oriented links in M quotiented by the Homflypt skein relations: (1) x --1L+ -- xL -- = (s -- s--1 )L0; (2) L with a positive twist = (xv--1)L; (3) L ⊔ O = u-u-1 s-s-1L where O is the unknot. We give two bases for the relative Homflypt skein module of the solid torus with 2 points in the boundary. The first basis is related to the monomial basis of S( S1 x D2) given by V. Turaev and also J. Hoste and M. Kidwell; the second basis is related to a Young idempotent basis for S(S 1 x D2) based on the work of A. Aiston, H. Morton and C. Blanchet. We prove that if the elements s2n -- 1, for n a nonzero integer, and the elements s2m -- upsilon 2, for any integer m, are invertible in k, then S(S1 x S2) = k-torsion module ⊕ k. Here the free part is generated by the empty link &phis;. In addition, if the elements s2m -- upsilon 4, for m an integer, are invertible in k , then S(S1 x S2) has no torsion. We also obtain some results for more general k.

#### Recommended Citation

Zhong, Jianyuan, "Homflypt Skein Modules." (2000). *LSU Historical Dissertations and Theses*. 7310.

https://digitalcommons.lsu.edu/gradschool_disstheses/7310

#### ISBN

9780599906372

#### Pages

57