The anti-diagonal filtration: reduced theory and applications
arXiv:1109.3425 · doi:10.1093/imrn/rnu177
Abstract
Given a knot K in S^3, Seidel and Smith described in arXiv:1002.2648v3 a graded cohomology group Kh_{symp,inv}(K), a variant of their symplectic Khovanov cohomology group. They also constructed a spectral sequence converging to the Heegaard Floer-hat homology group for the connected sum of the double branched cover and a copy of S^{2}xS^{1} (with E^1-page isomorphic to a direct summand of Kh_{symp,inv}(K)). In a previous paper (arXiv:1004.2476v5), we showed that the higher pages of this spectral sequence are knot invariants. Here we discuss a reduced version of the spectral sequence which directly computes HF-hat of the double branched cover. Under some degeneration conditions, one obtains a new absolute Maslov grading on that group. This occurs when K is a two-bridge knot, and we compute the grading in this case. We also extract some rational-valued knot invariants from this construction.
46 pages, 43 figures. Sequel to arXiv:1004.2476v5. Theorem numbering conventions changed. Theorem 1.0.3 (formerly Theorem 2) now only appears with field coefficients. Exposition in Sections 2 and 3 expanded. Changes made to the proof of Lemma 3.9.2, formerly Lemma 19. Section 3.10 added. This is the version to appear in Int. Math Res. Notices, published online Oct. 2014