Equivariant Khovanov-Rozansky Homology and Lee-Gornik Spectral Sequence

TL;DR

This paper establishes a formula linking equivariant sl(N) Khovanov-Rozansky homology with the Lee-Gornik spectral sequence, showing their equivalence and introducing a new link invariant called torsion width.

Contribution

It provides a new formula and an alternative definition for the Lee-Gornik spectral sequence, demonstrating their equivalence and introducing the torsion width invariant.

Findings

The Lee-Gornik spectral sequence can be derived from the equivariant homology decomposition.

Equivariant sl(N) Khovanov-Rozansky homology can be recovered from the spectral sequence.

Torsion width determines spectral sequence collapse and relates to homological thickness.

Abstract

Lobb observed in [arXiv:1103.1412] that each equivariant sl(N) Khovanov-Rozansky homology over C[a] 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 sl(N) Khovanov-Rozansky homology over C[a] 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 exterior algebra action on the sl(N) 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 sl(N) Khovanov-Rozansky homology. We use the torsion width to explain why the Lee spectral sequences of certain H-thick links collapse so fast.