On the Homflypt skein module of $S^1 times S^2$ |
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Authors: | Patrick M. Gilmer Jianyuan K. Zhong |
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Affiliation: | (1) Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA (e-mail: gilmer@math.lsu.edu) , US;(2) Program of Mathematics and Statistics, Louisiana Tech University, Ruston, LA 71272, USA (e-mail. kzhong@coes.LaTech.edu) , US |
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Abstract: | Let k be a subring of the field of rational functions in x, v,s which contains . If M is an oriented 3-manifold, let 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) ; (2) L with a positiv e twist ; (3) 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 basis of given by J. Hoste and M. Kidwell and also V. Turaev; the second basis is related to a Young idempotent basis for based on the work of A. Aiston, H. Morton and C. Blanchet. We prove that if the elements , for n a nonzero integer, and the elements , for any integer m, are invertible ink, then -torsion module . Here the free part is generated by the empty link . In addition, if the elements , for m an integer, are invertible in k, then has no torsion. We also obtain some results for more general k. Received January 7, 2000; in final form September 20, 2000 / Published online April 12, 2001 |
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