Equality of the dimensions of the solid-torus and annular Khovanov theories

Establish that the homology theory \(\mathcal{H}^{S^1\times D^2}(K,\,\cdot\,)\) constructed from dg-KLRW algebras and the Annular Khovanov Homology \(AKh(K,\,\cdot\,)\) have the same dimension for knots \(K\subset S^1\times D^2\).

Background

The paper constructs a homological invariant HS1×D2\mathcal{H}^{S^1\times D^2} for knots in the solid torus using dg-KLRW algebras with Verma-module-colored strands and fundamental-representation strands. The authors compare this invariant with Annular Khovanov Homology through explicit computations for the unknot and trefoil in S1×D2S^1\times D^2.

The examples suggest that the two theories are closely related, but the paper does not establish an isomorphism or even prove equality of dimensions in general. The conjecture asks for equality of dimensions as a general relationship between the two theories.

References

In Section \ref{sec:example}, we examine the homological invariant \Phi' in the case that the braid \beta is a single blue strand. This defines a homology theory of knots in S1\times D2, \mathcal{H}{S1\times D2}(\hspace{2mm}\cdot \hspace{2mm},) $, which we compare to Annular Khovanov Homology in several explicit examples. Short of an honest isomorphism between them, we show the two theories seem to at least have the same dimension.

— On Diagrammatic Categorification of Verma Modules I: Braiding  (2609.10941 - Guicardi, 10 Sep 2026) in Section 1, Introduction, paragraph beginning "The goal of this paper and its sequel"; reiterated in Section 6.2, Section "Unknot and Annular Khovanov Homology"