On the Kauffman skein modules

Details On the Kauffman skein modules On the Kauffman skein modules

2001-10-18

arxiv.org/abs/math/0110200v1

Manuscripta Mathematica, 109 (2002) 1, 29-47.

Mathematics

Geometric Topology

17 pages, 44 figures, amslatex, using epsf.tex

Scientific paper

Let k be a subring of the field of rational functions in \alpha, s which contains \alpha^{1}, \alpha^{-1}, s^{1}, 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}(\alpha, s), the field of rational functions in \alpha, 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(S^{1} \times S^2) is generated by the empty link, i.e., K(S^{1} \times S^2)=k.

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