10 problems

Let $L$ be a link in $S^3$. Write $coverline{H}(L)$ and $H^-(L)$ for the reduced, middle, and unreduced HOMFLY-PT homologies of $L$, and $coverline{\operatorname{HFK}}(L)$,…

Manolescu's conjecture. When $S$ is in braid position, the homology $H_{*}(C_{F}(S))$ is isomorphic to the HOMFLY-PT homology of $S$.

Manolescu's conjecture. There is an isomorphism

Let $L$ be a link in $S^3$. The HOMFLY-PT homologies carry gradings $\operatorname{gr}_T$ and $\operatorname{gr}_M$, and the grading-shifted knot Floer homologies…

Let $L$ be an arbitrary link, let $\mathrm{HY}(L)$ denote its homology, and let $\mathcal{E}_k$ and $\mathcal{F}_k$ be the operators acting on it. Let $\mathcal{H}_2$ be the Lie al…

Consider refined Chern-Simons theory with gauge group $\mathrm{SU}(N)$ and torus links, with colors labeled by rectangular Young diagrams. Let the large-$N$ limit refer to the limi…

Refined double-twist conjecture. The Poincare polynomial of $[r^s]$-colored HOMFLY-PT homology of $K_{m,n}$ is

Let $\overline{\mathrm{KhR}}(\mathcal{K})$ denote the reduced uncolored HOMFLY-PT homology of a knot or link $\mathcal{K}$, equipped with its $q$-grading. The q-symmetry conjecture…

The cube-filtration HOMFLY-PT conjecture. The page $E^{F}_{2}(D)$ is isomorphic to HOMFLY-PT homology.

Let $E_k(N)$ denote the spectral sequences from HOMFLY-PT homology to $sl(N)$ homology for $N\geq 1$. Rasmussen's conjecture. HOMFLY-PT homology should admit a symmetry that also g…