23 problems

Let $\mathcal{H}^{\mathrm{HOMFLY}}(K)$ denote HOMFLY homology of a knot $K$, and let $\mathcal{H}^{\mathfrak{sl}(N)}(K)$ denote its $\mathfrak{sl}(N)$ homology. Let $d_N$ be the co…

Let $K\subset S^3$ be a knot, and let $\mathcal{H}(K)$ be the triply graded homology theory associated with the HOMFLY polynomial. For each positive integer $N$, the source conside…

Let $K\subset S^3$ be a knot, let $\overline{H}_N(K)$ denote its $sl(N)$ KhovanovRozansky homology, and let $P_K(a,q)$ be its normalized HOMFLY polynomial. The triply graded HOMFLY…

HOMFLY homology differentials conjecture. There is a homology theory $\mathcal H_*$ categorifying the HOMFLY polynomial, with differentials $\{d_N\}$ satisfying those three axioms,…

A sequence of chain complexes of filtered graded vector spaces is said to be homologically $q$-holonomic when it satisfies the proposed notion of homological $q$-holonomicity. For…

Let $L$ be a link in the thickened annulus, and let $\mathcal{H}^{\mathrm{ann}}_{\mathrm{HOMFLY}}(L)$ denote its annular HOMFLY homology. Let…

For a link $L$, let ${\mathscr{H}}^{\rm fin}_{[r]}(L)$ be its finite-dimensional homology in the symmetric representation $[r]$. Define the $\delta$-grading on a generator $x$ by ……

Let $L$ be a link and let $\lambda$ be a coloring. Denote by ${\mathscr{H}}^{\rm fin}_\lambda(L)$ the finite-dimensional homology categorifying the finite colored HOMFLY invariant…

Let $\mathcal{H}^{\mathrm{HOMFLY}}(K)$ be uncolored HOMFLY homology of a thick knot $K$, and let $d_0$ be the HFK-like differential. HFK differential conjecture. The differential…

Let $K$ be a knot, and let $H^{\Lambda^r}_{i,j,*}(K)$ and $H^{S^r}_{i,j,*}(K)$ denote, respectively, its antisymmetric- and symmetric-coloured HOMFLY homology groups, with indices…

ORS geometric conjecture. The reduced HOMFLY homology of the knot link$(C,x)$ has Poincare polynomial

Filtration-grading conjecture. For torus knots, the filtration grading coincides with the grading induced by $\mathcal{F}^{\bullet}$ on $H_{m/n}$.

Stable homology conjecture. There is an isomorphism

DGR's differential conjecture. The group $\mathcal{H}_K$ should admit differentials $d_N$ for all $N\in\mathbb{Z}$, mutually anticommuting, such that for $N>0$

DGR's involution conjecture. The group $\mathcal{H}_K$ admits an involution generalizing the $q\mapsto q^{-1}$ symmetry of the HOMFLY polynomial.

Filtration coincidence conjecture. The three filtrations are the same:

Let $X_{mn+r,n}$ be the germ at the origin of $y^n=x^{mn+r}$, with $0<r<n$ and $(r,n)=1$, and let $\mu$ be its Milnor number. For a partition $\lambda$ of $n$, use the arm, leg, co…

Let $X_{k,n}$ be the germ at the origin of $y^n=x^k$, where $(k,n)$ are coprime, and let $\mu$ be its Milnor number. Let $F_{k/n}$ be a $\mathbb{C}^*\times\mathbb{C}^*$-equivariant…

Let $X_{k,n}$ be the germ at the origin of $y^n=x^k$, with coprime positive integers $k,n$, and let $\mu$ be its Milnor number. Set $T_1=q^2$ and $T_2=1/(q^2t^2)$. Let $eL_{k/n}$ b…

For the stable torus-knot link $T(n,\infty)$, let $\overline{\mathscr{P}}$ denote its unreduced HOMFLY-homology Poincaré polynomial, with variables $a,q,t$. Dunfield–Gukov–Rasmusse…

Let $X$ be the germ of a plane curve singularity with Milnor number $\mu$. Let $X^{[l\le l+m]}$ be the incidence variety of pairs of ideals $(I,J)$ with…

Let $C$ be a plane curve singularity, let $J(C)$ denote its compactified Jacobian, and consider the cohomology of $J(C)$ equipped with a certain filtration. Compactified-Jacobian f…

For $q=p+1$, the construction of generators for HOMFLY homology counts Schröder paths in a square. The proposed generators carry gradings, and the associated bigraded Poincaré poly…