28 problems

Let $\mathcal H^{daha}$ denote the DAHA superpolynomial and let $KhR$ denote the reduced colored Khovanov–Rozansky polynomial of an iterated torus link. DAHA–KhR coincidence conjec…

Let $\mathbf H(q,t,a)$ be the uncolored link series, and let $\kappa$ be the number of components of the algebraic link. Weak Riemann hypothesis. At $a=0$, $\mathbf H(q,t,0)$ has o…

Let $\mathcal H^{daha}(q,t,a)$ be the DAHA superpolynomial of an algebraic link, colored by rectangle Young diagrams, and let $\kappa$ be the number of link components. Positivity…

Let $\mathcal{J}_\ell$ be the moduli space of flags of modules associated with an algebraic knot, and let its geometric superpolynomial be defined using the Borel–Moore homology of…

Let $\mathcal{L}$ and $'\!\mathcal{L}$ be algebraic links with coloring partitions $\lambda^j$ and $'\!\lambda^j$, and let their iterated-torus data be…

Let $\mathcal{R}$ be a plane-curve-singularity ring with branches indexed by $1\le i\le\kappa$, and let $\sigma=(c_1\ge\cdots\ge c_\kappa>0)$. Define the motivic superpolynomial ……

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…

Let $\scD^{(1)}_{\ell}$ be the $\ell$-dimensional module singled out among the four modules in the shortening regime, with basis indexed by $0\leq j\leq \ell-1$. Let…

Let $q$ and $t$ satisfy the shortening condition defining the $(k+1)$-dimensional module $\repV_{k+1}$ of the spherical double affine Hecke algebra, with basis indexed by…

Let $N$ be a positive integer and define … Let $\overline{P}_N(q_u,x_u,x_d;\boldsymbol{c})$ be the reduced DAHA polynomial, let $\mathbf{s}_N(x)$ and $\mathbf{B}_N(q)$ be the basis…

Let $N$ be a positive integer, let $q_u=q^{\frac{1}{2}}$, and let $S_{N-1}(\mathcal{A}_{(1,2)})$ be the indicated Chebyshev-type expression in the DAHA operator…

Let $\boldsymbol{c}$ be a simple closed curve on the genus-two surface $Sigma_{2,0}$, and let $\overline{P}_n(q_u,x_u,x_d;\boldsymbol{c})$ denote its reduced DAHA invariant. The pa…

Singular superpolynomial conjecture. For all $i,d\geq0$, $H^{\!BM}_{2i+1}(\mathcal{J}^{\ell}_{\mathcal{R},{\mathsf \varrho}}[d])$ vanishes, and

Stable Khovanov–Rozansky conjecture. After this substitution,

Main conjecture. For all $i,d\geq0$, the relative homology $H_{2i+1}(\overline{J}^m_{\mathcal{R}}[d],\overline{J}^m_{\mathcal{R}}[d+1])$ vanishes, and

Rectangular DAHA-superpolynomial positivity conjecture. The coefficients of $H\!D_{r,s}(j\omega_i;q,t,a)$ are positive integers. After the displayed transformation, the resulting p…

Cherednik's DAHA–Jones conjecture. Upon specializing $t\mapsto q$, the DAHA-Jones polynomial agrees with the corresponding quantum invariant up to this overall factor:

Let $T^{{\mathsf r},{\mathsf s}}$ be a torus knot with ${\mathsf r}>{\mathsf s}$, and let $\lambda$ and $\mu$ be Young diagrams. Write…

Let $\mathcal{H}$ and $\mathcal{H}^{\checkmark}$ denote the two DAHA-type algebras considered in the paper, and let $\mathcal{B}_q$ be the associated braid-group representation. Co…

Positivity conjecture. The coefficients of $\overline{E}_b^\dag$ for all $b\in B$ belong to $\mathbf{Z}_+[q^{-1}]$.

Let $K=K_{r,s}$ be a knot with $r,s\in\mathbf{Z}$, and let $b$ be a dominant weight for all $B_m$ with $m\ge n\ge2$. Type-B hyper-polynomial conjecture. There exists a polynomial…

Let $K=K_{r,s}$ be a knot with $r,s\in\mathbf{Z}$, and let $b=\sum_{i=1}^{n}c_i\epsilon_i$ be dominant for all $C_m$ with $m\ge n\ge2$. Type-C hyper-polynomial conjecture. There ex…

Let $\mathcal{H}^{C}_{r,s}(b;q,t,u,a)$ denote the type-$C$ hyper-polynomial and let $\widetilde{J\!D}^{C_m}_{r,s}$ be the corresponding type-$C_m$ DAHA-Jones polynomial. Existence…

Let $H\!D_{r,s}(b;q,t,a)$ have positive coefficients, let $\mathcal{H}=\bigoplus\mathcal{H}_{i,j,k}$ be the triply graded space for the torus knot $\{r,s\}$ and weight $b$, and let…

Let $b\in P_+$ correspond to a Young diagram $\mu_b$ with rows of lengths $m_i$, and let $b_i=m_i\omega_1$. Write $\operatorname{ord}$ for the order of $\mu_b$. Evaluation conjectu…