Generalized Skein Modules of Surfaces

TL;DR

This paper explores a broad class of marked cobordism categories related to generalized sl(n) link homology theories, analyzing their skein modules within 3-manifolds and providing explicit presentations for these modules.

Contribution

It introduces a large family of marked cobordism categories for generalized sl(n) link homology and derives explicit presentations for their skein modules.

Findings

Explicit presentations for several generalized skein modules.

Identification of a large family of relevant cobordism categories.

Enhanced understanding of skein modules in 3-manifolds.

Abstract

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.