A $y$-ification of Khovanov homology

Motivated by the $y$-ification of HOMFLY–PT homology by Gorsky and Hogancamp, and the ${\mathfrak{s}\mathfrak{l}}_2$-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 ${\mathfrak{s}\mathfrak{l}}_2$ 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.