A deformation of Asaeda–Przytycki–Sikorahomology

We define a 1-parameter family of homology invariants for links in thickened oriented surfaces.It recovers the homology invariant of Asaeda, Przytycki and Sikora (Algebr.Geom.Topol. 4 (2004Topol.4 ( ) 1177Topol.4 ( -1210) ) and the invariant defined by Winkeler (Michigan Math.J. 74 (2024) 1-31).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 nontrivial 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 our previous work (Selecta Math.29 (2023) art.id.84).