Abstract: I'll describe new invariants of a framed link in a 3-manifold, which arise as the pages of a spectral sequence generalizing the surgery exact triangle in monopole Floer homology. The construction introduces a connection between the topology of link surgeries and the combinatorics of polytopes called graph associahedra. For a classical link L in S³, we obtain a sequence of bigraded vector spaces, interpolating between the reduced, Z/2Z Khovanov homology of L and a version of the monopole Floer homology of the branched double cover. In addition, this perspective yields a simple, topological proof that odd Khovanov homology is mutation invariant. I'll emphasize low-dimensional topology through lots of pictures, and not the technical details of Floer homology.

Paper reference: arxiv.org/abs/0903.3746, arxiv.org/abs/0909.0816