A $y$-ification of Khovanov homology

TL;DR

This paper introduces a new $y$-ification framework for Khovanov homology, compatible with existing theories, enabling the distinction of knots with identical traditional invariants through diagrammatic and algorithmic methods.

Contribution

It constructs $y$-ifications of Khovanov homology within Bar-Natan's framework and defines an $rak{sl}_2$-action, providing a new tool to distinguish knots with identical classical invariants.

Findings

Successfully distinguishes the Conway and Kinoshita--Terasaka knots

Provides an elementary, diagrammatic approach to $y$-ification

Ensures compatibility with Rasmussen's spectral sequence

Abstract

Motivated by the $y$-ification of HOMFLY--PT homology by Gorsky and Hogancamp, and the $\mathfrak{sl}_2$-action of Gorsky, Hogancamp, and Mellit, we construct $y$-ifications of Khovanov homology and its equivariant versions within Bar-Natan's framework for tangles, and define an action of the element $e$ in $\mathfrak{sl}_2$ on these $y$-ifications. We then prove that our construction is compatible with the previous ones under Rasmussen's spectral sequence from HOMFLY--PT homology to Khovanov homology. Our construction is elementary and well suited to diagrammatic manipulations and algorithmic implementations. As a result, we verify directly that these additional structures distinguish pairs of knots with identical Khovanov homology and HOMFLY--PT homology, in particular the Conway knot and the Kinoshita--Terasaka knot.