Stable deformed $\mathfrak{gl}_N$ homology of torus knots

TL;DR

This paper computes the $E_2$ page of a spectral sequence for $ ext{gl}_N$ homology of torus knots, confirming a conjecture and explicitly calculating the homology using link-splitting deformation techniques.

Contribution

It introduces the use of $y$-ification to explicitly compute the $y$-ified $ ext{gl}_N$ stable homology of torus knots for all $N$, confirming a conjecture.

Findings

Computed the $E_2$ page of the Rasmussen spectral sequence for torus knots.

Confirmed a weak form of a conjecture by Oblomkov, Rasmussen, and others.

Explicitly calculated the $y$-ified $ ext{gl}_N$ homology for all $N$.

Abstract

We compute the $E_2$ page in the Rasmussen spectral sequence from triply graded to $\mathfrak{gl}_N$ Khovanov--Rozansky stable homology of torus knots. This confirms a weak form of the conjecture of the second author, Oblomkov, and Rasmussen. The main tool is the link-splitting deformation, or $y$-ification, of link homology; in the $y$-ified context, the relevant Rasmussen spectral sequence collapses and we explicitly compute the $y$-ified $\mathfrak{gl}_N$ stable Khovanov--Rozansky homology of torus knots for all $N$.