Equivariant Khovanov–Rozansky homology and Lee–Gornik spectral sequence

  • Hao Wu

    George Washington University, USA

Abstract

Lobb observed in [8] that each equivariant Khovanov–Rozansky homology over admits a standard decomposition of a simple form.

In the present paper, we derive a formula for the corresponding Lee–Gornik spectral sequence in terms of this decomposition. Based on this formula, we give a simple alternative definition of the Lee–Gornik spectral sequence using exact couples. We also demonstrate that an equivariant Khovanov–Rozansky homology over can be recovered from the corresponding Lee–Gornik spectral sequence via this formula. Therefore, these two algebraic invariants are equivalent and contain the same information about the link.

As a byproduct of the exact couple construction, we generalize Lee's endomorphism on the rational Khovanov homology to a natural -action on the Khovanov–Rozansky homology.

A numerical link invariant called torsion width comes up naturally in our work. It determines when the corresponding Lee–Gornik spectral sequence collapses and is bounded from above by the homological thickness of the Khovanov–Rozansky homology. We use the torsion width to explain why the Lee spectral sequences of certain H-thick links collapse so fast.

Cite this article

Hao Wu, Equivariant Khovanov–Rozansky homology and Lee–Gornik spectral sequence. Quantum Topol. 6 (2015), no. 4, pp. 515–607

DOI 10.4171/QT/70