On the Kauffman skein modules

2001/10/18 by Jianyuan K. Zhong, Bin Lu
Mathematics · #math.GT #math.QA #msc:57M

paper · pdf

published as Manuscripta Mathematica, 109 (2002) 1, 29-47. · 17 pages, 44 figures, amslatex, using epsf.tex

arxiv created 2001/10/18 · arxiv updated 2009/11/30

Abstract

Let k be a subring of the field of rational functions in α, s which contains α¹, α^-1, s¹, s^-1, . Let M be a compact oriented 3-manifold, and let K(M) denote the Kauffman skein module of M over k. Then K(M) is the free k-module generated by isotopy classes of framed links in M modulo the Kauffman skein relations. In the case of k=Q(α, s), the field of rational functions in α, s, we give a basis for the Kauffman skein module of the solid torus and a basis for the relative Kauffman skein module of the solid torus with two points on the boundary. We then show that K(S¹ × S²) is generated by the empty link, i.e., K(S¹ × S²)=k.

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