International Journal of Mathematics and Mathematical Sciences
Volume 32 (2002), Issue 9, Pages 515-554
doi:10.1155/S016117120201181X

Skein modules of links in cylinders over surfaces

Abstract

We define the Conway skein module 𝒞 (M) of ordered based links in a 3-manifold M. This module gives rise to 𝒞 (M)-valued invariants of usual links in M. We determine a basis of the ℤ[z]-module 𝒞 (Σ×[0,1])/Tor (𝒞 (Σ×[0,1])), where Σ is the real projective plane or a surface with boundary. For cylinders over the Möbius strip or the projective plane, we derive special properties of the Conway skein module, among them a refinement of a theorem of Hartley and Kawauchi about the Conway polynomial of strongly positive amphicheiral knots in S3. In addition, we determine the Homfly and Kauffman skein modules of Σ×[0,1] where Σ is an oriented surface with boundary.