18 problems

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 $\mathcal{H}^{mot}(q,t,a)$ be the motivic superpolynomial of a plane curve singularity, and let $L(q,t,a)$ denote the corresponding generalized Galkin–Stöhr $L$-function. Secon…

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 ……

Generalized colored superpolynomial conjecture. The superpolynomial colored by a symmetric representation has the form

t-deformed GM-invariant quiver correspondence. The invariant can be written as

The (a,t)-deformed $F_K$ conjecture. There should exist a four-variable function satisfying

Superpolynomial full twist conjecture. We conjecture that

Let $\mathcal{P}_{n,m}$ denote the superpolynomial of the $(n,m)$ torus knot, and let $\mathcal{P}_{\mathrm{unknot}}^{[m]}$ denote the superpolynomial of the unknot colored in the…

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…

Let $Y_{m/n}$ be the set of $m/n$ Dyck paths, let $v(D)$ be the set of internal vertices of $D$, and let $\beta(P)$ be the stated vertex statistic. Let…

Let $\mathcal{P}_{n,m}^{\lambda}(u,q,t)$ be the stabilized superpolynomial and define its reduced version by … Cherednik's duality conjecture. For the transposed partition…

Let $\mathcal{P}_{n,m}^{\lambda,N}(q,t)$ be the DAHA superpolynomial associated with a partition $\lambda$ in rank $N$. Cherednik's stabilization conjecture. There exists a polynom…

Let $m$ and $n$ be coprime positive integers, let $\mathcal{P}_{n,m}(u,q,t)$ denote the superpolynomial, and let $D$ range over lattice paths below the main diagonal in an…

Let $K$ be a knot, let ${\cal P}_n(K;a,q,t)$ be its colored superpolynomial, and let $a_0,\ldots,a_k$ denote the coefficients of a recurrence operator, with $k$ its order. The quan…

Generalized volume conjecture for colored superpolynomials. In this limit, the large-color behavior is

Let $K$ be a knot, let $S^r$ and $\Lambda^r$ denote the rank-$r$ symmetric and antisymmetric representations, respectively, and let $\square$ denote the fundamental representation.…

Let $\mathbf{\Lambda}_N^{\theta}=\mathbb C[x_1,\dots,x_N,\theta_1,\dots,\theta_N]^{S_N}$ be the ring of symmetric superpolynomials, and let $\mathcal I_N^{(k,r)}$ be the span of Ja…

Let $L$ be the Hopf link colored by representations $lambda$ and $mu$. For each integer $N$, let $\overline{\cal P}_{\mathfrak{sl}(N); \lambda, \mu}(\mathbf q,\mathbf t)$ denote th…