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.
Khovanov's categorification of the Jones polynomial [Kho00], known as Khovanov homology, assigns to an oriented link L ⊂ S3 a bigraded homology Kh * , * (L; Q) whose graded Euler characteristic recovers the Jones polynomial. A persistent theme in the subject is that adding extra structures on Khovanov homology strengthens the invariant, and often detects phenomena invisible at the level of polynomials or bigraded vector spaces. A foundational instance is the work of Kronheimer and Mrowka showing that Khovanov homology detects the unknot, a property still unknown for the Jones polynomial [KM11]. The result is obtained by relating Khovanov homology with singular instanton Floer homology via a spectral sequence. Subsequent works also show that Khovanov homology detects the unlink, a property that does not hold for the Jones polynomial [HN13; BS15]. 12 Another influential direction was led by Lee's deformation of Khovanov homology [Lee05], which gave Rasmussen's concordance invariant s and its applications to slice genus bounds [Ras10]. Subsequently, Bar-Natan reformulated Khovanov homology via tangles and cobordisms [Bar05], and Khovanov unified rank-two deformations of Khovanov homology via Frobenius extensions, yielding the equivariant versions of Khovanov homology [Kho06]. Their algebraic structures have been studied in [Nao06; Tur06; MTV07; Shu14; Wig16; Tur20; KR22; KS25a; CY25]. In recent works, the torsion part of equivariant Khovanov homology has found topological applications, as in [Ali19; AD19; Sar20; Guj20; Cap+21; Zhu22; Hay23; LMZ24; ILM25]. Up-to-sign functoriality with respect to link cobordisms was established in [Jac04; Bar05], 3 and was applied to the detection of exotic slice disks [HS24;Hay23]. Lipshitz and Sarkar gave a space-level refinement of Khovanov homology, which equips it with Steenrod squares and strictly refines the s-invariant [LS14a; LS14c;LS14b]. There are also variants adapted to additional geometry or symmetry of the link, as in [APS04; GLW18; Pol19; LW21; LS24; San25 ;Bor+25]. These developments underscore the potential of the algebraic structures that Khovanov homology admits, and the geometric properties they reflect.
In this paper, we focus on the works of Gorsky and Hogancamp [GH22] and of Gorsky, Hogancamp and Mellit [GHM24], which are extensions of HOMFLY-PT homology-a triply graded link homology theory introduced by Khovanov and Rozansky as a categorification of the HOMFLY-PT polynomial [KR08b]. In [GH22], Gorsky and Hogancamp gave a y-ification of HOMFLY-PT homology by adjoining formal parameters y c for each component c of a link L, giving a triply graded module over Q[x c , y c ] c∈π0(L) . This extension is designed to give a link-splitting spectral sequence analogous to the one given by Batson and Seed on Khovanov homology [BS15], while also fitting into conjectural geometric structures coming from Hilbert schemes of points in the plane. Building on this construction, in [GHM24], Gorsky, Hogancamp and Mellit further construct a family of commuting endomorphisms F k on the y-ified HOMFLY-PT homology, analogous to the action of tautological classes in the cohomology of character varieties. They prove a hard Lefschetz-type statement for the second operator F 2 , which in turn extends to an sl 2 -action on the homology group, further proving the Q ↔ T Q -1 symmetry conjectured by Dunfield-Gukov-Rasmussen [DGR06].
The purpose of this paper is to bring the idea of y-ifications to Khovanov homology and to study the resulting structure both theoretically and computationally. We construct a framework independent of the previous work, and then relate the two formulations via Rasmussen’s spectral sequence from HOMFLY-PT homology to Khovanov homology [Ras15].
We start by explaining the construction of the y-ified Khovanov complex y[β] for a braid diagram β on n strands. Here, for simplicity, we work in the non-equivariant setting over Q. Let B n ⊂ ∂I 2 denote the set of boundary points of β, with n incoming points {P i } on I × {0} and n outgoing points {P ′ i } on I × {1} (see Figure 1a), and w ∈ S n the underlying permutation of β. Regard β as a tangle diagram with boundary B n and consider the formal Khovanov bracket [β] of β-a complex in the additive category of dotted cobordisms Cob(B n ) defined in [Bar05]. We follow the grading convention that a dot has positive quantum degree 2, which is opposite from [Kho00; Bar05] and the same as [Kho06;KR08a], since it better suits with the grading convention for HOMFLY-PT homology.
For any regular point p on β (i.e. a point which is not a crossing), a bidegree (0, 2) endomorphism X p of [β] is given by an array of identity cobordisms each with a dot placed on the component that contains p. Let β 0 be the orientation preserving resolution of β, which is given by n vertical arcs. We define a bidegree (0, 2) endomorphism x p of [β] by
x p := X p if p lies on the odd indexed strand of β 0 , -X p otherwise. This sign choice is
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