Generalized Skein Modules of Surfaces

Frobenius extensions play a central role in the link homology theories based upon the sl(n) link variants, and each of these Frobenius extensions may be recast geometrically via a category of marked cobordisms in the manner of Bar-Natan. Here we explore a large family of such marked cobordism categories that are relevant to generalized sl(n) link homology theories. We also investigate the skein modules that result from embedding these marked cobordisms within 3-manifolds, and arrive at an explicit presentation for several of these generalized skein modules.