11 problems

Refined skein recursion conjecture. The equation obtained for the refined Chern–Simons model is the correct refinement of the skein recursion relations for the unknot.

Let $T_{m,n}$ be a torus knot, and let $\bar{\operatorname{rCS}}_{\lambda}(T_{m,n};a,q,t)$ denote its stable large $N$ refined Chern–Simons invariant, where the color $\lambda$ is…

Refined torus skein conjecture. The refinement of the skein algebra of the torus is the quantum toroidal $cmathfrak{gl}_1$ algebra.

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 $T_{m,n}$ be a torus link with $L=\operatorname{gcd}(m,n)$ components. Let $\mathcal{M}_{g,h,\beta}$ be the relevant moduli space of holomorphic curves, and let…

Conjectural generating-function claim. For $p>1$, $\widetilde{{\cal Z}}_N^{\, {\rm BH}}(\phi_0,\phi_2,y;p)$ is the corresponding generating function for the $\chi_y$-genus of the i…

Let $N$ and $k$ be positive integers denoting the rank and level of refined Chern–Simons theory. For a link in $S^3$, let the associated three-variable Laurent polynomial be expres…

Fundamental-representation vanishing R-charge conjecture. For Wilson lines in the fundamental representation $R=\tableau{1}$, the states contributing to the refined Chern–Simons in…

Finite-N and large-N refined index conjecture. The refined index is computed by refined Chern–Simons theory: for $SL_N$ homologies, by finite-$N$ refined Chern–Simons theory, and f…

Fundamental-representation superpolynomial conjecture. The invariant ${\cal P}_{\tableau{1}}(K)({\bf q},{\bf t},{\bf a})$ computes the superpolynomial of the HOMFLY knot homology.

Refined Chern–Simons index conjecture. The refined index is computed by refined $SU(N)$ Chern–Simons theory.