A deformation of Asaeda-Przytycki-Sikora homology

Abstract: We define a 1-parameter family of homology invariants for links in thickened oriented surfaces. It recovers the homology invariant of Asaeda-Przytycki-Sikora (arxiv:0409414) and the invariant defined by Winkeler (arxiv:2106.03834). The new invariant can be regarded as a deformation of Asaeda-Przytycki-Sikora homology; it is not a Lee-type deformation as the deformation is only non-trivial when the surface is not simply connected. Our construction is motivated by computations in singular instanton Floer homology. We also prove a detection property for the new invariant, which is a stronger result than the main theorem of arxiv:2208.13963.