The $H_{1-1}$–$\mathit{HFK}_{2}$ identification conjecture

Let $K$ be a knot in $S^{3}$. Let $C_{1\pm1}(D)$ be the cube-of-resolutions complex associated to a plat braid diagram $D$ for $K$, with differential $d=d_{0}+d_{1}$, and define its total homology by

$$H_{1-1}(K)=H_{*}(C_{1\pm1}(D),d_{0}+d_{1}).$$

Let $\mathit{HFK}_{2}(K)$ denote the singly graded homology theory defined by counting holomorphic discs through the additional basepoints.

$H_{1-1}$–$\mathit{HFK}_{2}$ conjecture. The total homology $H_{1-1}(K)$ is isomorphic to $\mathit{HFK}_{2}(K)$.

Together with the stated identification of reduced $\mathit{HFK}_{2}$ with $\delta$-graded knot Floer homology, this would identify the total homology of the algebraic cube-of-resolutions complex with the proposed intermediate knot homology and support the desired spectral-sequence picture. The paper has proved that $H_{1-1}(K)$ is a link invariant, but this isomorphism is presented as conjectural.