Volume 1, issue 1 (2001)

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The Homflypt skein module of a connected sum of 3–manifolds

Patrick M Gilmer and Jianyuan K Zhong

Algebraic & Geometric Topology 1 (2001) 605–625
 arXiv: math.GT/0012056
Abstract

If $M$ is an oriented $3$–manifold, let $S\left(M\right)$ denote the Homflypt skein module of $M.$ We show that $S\left({M}_{1}#{M}_{2}\right)$ is isomorphic to $S\left({M}_{1}\right)\otimes S\left({M}_{2}\right)$ modulo torsion. In fact, we show that $S\left({M}_{1}#{M}_{2}\right)$ is isomorphic to $S\left({M}_{1}\right)\otimes S\left({M}_{2}\right)$ if we are working over a certain localized ring. We show the similar result holds for relative skein modules. If $M$ contains a separating $2$–sphere, we give conditions under which certain relative skein modules of $M$ vanish over specified localized rings.

Keywords
Young diagrams, relative skein module, Hecke algebra
Primary: 57M25