Twisted skein homology

Abstract

We apply the techniques of totally twisted Khovanov homology to Asaeda, Przytycki, and Sikora's construction of Khovanov type homologies for links and tangles in I-bundles over (orientable) surfaces. As a result we describe a chain complex built out of resolutions with only noncontractible circles whose homology is an invariant of the tangle. We use these to understand the δ-graded homology for links with alternating diagrams in the surface.