Knot homologies in monopole and instanton theories via sutures (Q2673136)

There are two approaches to Floer homology of a 3-manifold \(Y\), the monopole/instanton one going back to Floer himself and simplified by Kronheimer and Mrowka, and Heegard's one developed by Ozsváth and Szabó. When relativized to pairs \((Y,K)\), where \(K\subset Y\) is a knot, Heegard Floer homology comes in four definitional versions, but some of their monopole/instanton counterparts are currently unknown. In this paper, the author constructs candidates for the minus version of Heegard Floer homology based on sutured homology of Kronheimer and Mrowka when \(K\) is a based null-homologous knot (an extension to torsion-class knots is unproblematic). The proof is inspired by the construction of \textit{J. B. Etnyre} et al. [Geom. Topol. 21, No. 3, 1469--1582 (2017; Zbl 1420.57035)] that uses a sequence of balanced sutured manifolds in the Heegard setting. Unlike Kutluhan's holonomy filtration construction of the to-version [\textit{Ç. Kutluhan}, in: Proceedings of the 19th Gökova geometry-topology conference, Gökova, Turkey, May 28 -- June 2, 2012. Somerville, MA: International Press; Gökova: Gökova Geometry-Topology Conferences. 1--42 (2013; Zbl 1306.57003)], which heavily relied on monopole specific techniques, the author's one is more geometric and is carried out uniformly in monopole and instanton settings. The homology is defined with values in the mod-2 Novikov ring, and a Seifert surface of the knot induces a \(\mathbb{Z}\)-grading on it. There is a \(U\)-map that lowers the degree by \(1\), and becomes an isomorphism for \(i<N_0\) with some \(N_0\). When \(Y=S^3\) and the Seifert surface realizes the genus \(g\) of \(K\) the degree-\(g\) homology group is proved to be non-trivial. A surgery formula relating the minus homology of a knot to that of its dual is also proved, by performing a Dehn surgery of large enough slope along the knot. The \(U\)-module structure of this Floer homology was later studied in the author's joint work with Ghosh and Wong (Tau invariants in monopole and instanton theories), where it was computed for all twist knots.