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
Mathematisches Institut, Universität Basel, Rheinsprung 21, Basel CH-4051, Switzerland
Received 3 February 2001; Revised 18 December 2001
Copyright © 2002 Jens Lieberum. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.