2009/09/04 by Jonathan M. Bloom, Bloom, Jonathan M. · 3 citations
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AT #math.CO #math.GT #math.QA #math.SG
paper · pdf · doi:10.48550/arxiv.0909.0816
76 pages, 26 figures (v2: added 4 figures and Section 8.1 on realizations. v3: added table of figures, modified intro and figures 20 and 25)
arxiv created 2009/11/20 · arxiv updated 2009/12/08
To a link L in the 3-sphere, we associate a spectral sequence whose E2 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 Ek for k > 1 depend only on the mutation equivalence class of L. We define a mod 2 grading on the spectral sequence which interpolates between the delta-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.