Results on spectral sequences for monopole and singular instanton Floer homologies

Abstract

We study two gauge-theoretic Floer homologies associated to links, the singular instanton Floer homology introduced in [15] and the monopole Floer homology, which is explained in the book [16]. For both cases, we study in particular the spectral sequence that relates the Floer homologies to the Khovanov homologies of links. In our study of singular instanton Floer homology, we introduce a version of Khovanov homology for alternating links with marking data, W, inspired by singular instanton theory. We show that the analogue of the spectral sequence from Khovanov homology to singular instanton homology introduced in [15] for this marked Khovanov homology collapses on the E2 page for alternating links. We moreover show that for non-split links the Khovanov homology we introduce for alternating links does not depend on w; thus, the instanton homology also does not depend on W for non-split alternating links. 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 w. In our study of monopole Floer homology, we construct families of metrics on the cobordisms that are used to construct differentials in the spectral sequence relating the Khovanov homology of a link to the monopole Floer homology of its double branched cover, such that each metric has positive scalar curvature. This allows us to conclude that the Seiberg-Witten equations for these families of metrics have no irreducible solutions, so the differentials in the spectral sequence can be computed from counting only the reducible solutions.