A link surgery spectral sequence in monopole Floer homology

By J. Bloom. 2009

79 citations recorded by the source corpus.

To a link L in the 3-sphere, 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 > 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.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/ca89789023ee8922bce7f6219e23b4ee8f66134c

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Citations in the corpus

Khovanov homology is an unknot-detector (2010, background): “It is the intersection of S20 with a 4-ball arising as the regular neighborhood of [1, 3]×δ.”
Instantons and odd Khovanov homology (2014, methodology): “C = ( C(1) ⊗ C(2) ) ⊕ ( C(1)[3]⊗ C(2) ) ⊕ ( C(1) ⊗R ) ⊕ ( R⊗ C(2) )”
On the spectral sequence from Khovanov homology to Heegaard Floer homology (2008, background): “There is no direct analogue of a multi-diagram for the corresponding spectral sequence in monopole Floer homology (see [5]).”
Proof of the Arnold chord conjecture in three dimensions, II (2010, unknown): “Y0, then the induced map on Seiberg-Witten Floer cohomology with Z/2 coeﬃcients, HMd ∗ (X) : HMd ∗ (Y1) −→ HMd ∗ (Y0), (1.11) is not an isomorphism. Proof. It follows from [12, Thm. 42.2.1], see also [3, 13], that there is an exact triangle ··· −→ HMd ∗ (Y2) −→ HMd ∗ (Y1) HMd ∗ (X) −→ HMd ∗ (Y0) −→ ”
Odd Khovanov homology is mutation invariant (2009, methodology): “We elaborate on this perspective, and its relationship to the monopole Floer and Donaldson TQFT’s, in [5].”