A link surgery spectral sequence in monopole Floer homology

Abstract

To a link L ⊂ S 3 , we associate a spectral sequence whose E 2 page is the reduced Khovanov homology of L and which converges to a version of the monopole Floer homology of the branched double cover. The pages E k for k ⩾ 2 depend only on the mutation equivalence class of L. We define a mod 2 grading on the spectral sequence which interpolates between the δ-grading on Khovanov homology and the mod 2 grading on Floer homology. We also derive a new formula for link signature that is well adapted to Khovanov homology. More generally, we construct new bigraded invariants of a framed link in a 3-manifold as the pages of a spectral sequence modeled on the surgery exact triangle. The differentials count monopoles over families of metrics parameterized by permutohedra. We utilize a connection between the topology of link surgeries and the combinatorics of graph-associahedra. This also yields simple realizations of permutohedra and associahedra, as refinements of hypercubes.