The anti-diagonal filtration: reduced theory and applications

TL;DR

This paper introduces a reduced spectral sequence related to symplectic Khovanov cohomology that computes Heegaard Floer homology of double branched covers, providing new invariants and gradings for knots, especially two-bridge knots.

Contribution

It develops a reduced spectral sequence for Heegaard Floer homology, offering explicit gradings and invariants, particularly for two-bridge knots, advancing knot theory and Floer homology.

Findings

Reduced spectral sequence computes HF-hat of double branched cover.

New absolute Maslov grading under degeneration conditions.

Rational-valued knot invariants derived from the construction.

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.