Equivariant Khovanov–Rozansky homology and Lee–Gornik spectral sequence

Abstract

Lobb observed in [8] that each equivariant $\mathfrak{sl}(N)$ Khovanov–Rozansky homology over $\mathbb{C}[a]$ 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 $\mathfrak{sl}(N)$ Khovanov–Rozansky homology over $\mathbb{C}[a]$ 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 ${\bigwedge }^{\ast }{\mathbb{C}}^{N-1}$-action on the $\mathfrak{s}l(N)$ 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 $\mathfrak{sl}(N)$ Khovanov–Rozansky homology. We use the torsion width to explain why the Lee spectral sequences of certain H-thick links collapse so fast.