Filtrations on instanton homology

Abstract

The free abelian group C carries both a cohomological grading h and a quantum grading q. The differential dKh increases h by 1 and preserves q, so that the Khovanov cohomology is bigraded. We write F C for the decreasing filtration defined by the bigrading, so F C is generated by elements whose cohomological grading is not less than i and whose quantum grading is not less than j . In general, given abelian groups with a decreasing filtration indexed by Z Z, we will say that a group homomorphism has order .s; t/ if .F i;j / F iCs;jCt . So dKh has order .1; 0/. In [5], a new invariant I .K/ was defined using singular instantons, and it was shown that I .K/is related to Kh.K/ through a spectral sequence. The notation K here denotes the mirror image of K. Building on the results of [5], we establish the following theorem in this paper.