Tensor-factor relation for arbitrary annular knots

Prove that for every annular knot \(K\subset S^1\times D^2\) representing the class \([K]=n\in\pi_1(S^1\times D^2)\), and for an unknot \(U_n\) winding \(n\) times around the nontrivial cycle, the relation \(\mathcal{H}^{S^1\times D^2}(K,\,\cdot\,)\otimes AKh(U_n,\,\cdot\,)\cong AKh(K,\,\cdot\,)\otimes\mathcal{H}^{S^1\times D^2}(U_n,\,\cdot\,)\) holds.

Background

The paper verifies analogous tensor-factor identities in several examples, including the unknot and a trefoil in the solid torus. It also derives a related formula for a class of annular knots from comparisons with ordinary Khovanov homology.

The authors then propose that the same pattern should hold for any annular knot, with the winding number encoded by the reference unknot $U_n$. This statement remains conjectural and is presented as a generalization of the computed examples.