Khovanov homology and binary dihedral representations for marked links

Abstract

We introduce a version of Khovanov homology for alternating links with marking data, ω, inspired by instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology introduced in [10] for this marked Khovanov homology collapses on the E 2 page for alternating links. We moreover show that for non-split links the Khovanov homology we introduce for alternating links does not depend on ω; thus, the instanton homology also does not depend on ω for non-split alternating links.Finally, we study a version of binary dihedral representations for links with markings, and show that for links of non-zero determinant, this also does not depend on ω.