  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.
