12 problems

Let $K\subset S^3$ be a knot, let $U$ be the holonomy of the $U(N)$ Chern–Simons gauge field around $K$, and let $V$ be a source whose traces are summed over representations $r$. W…

Let $K$ be a knot and let $\overline{P}_r(a,q)$ denote its unreduced $S^r$-colored HOMFLY-PT polynomial. Let $Q_K$ be a symmetric quiver with $m$ nodes, adjacency matrix…

Let $K$ be a knot, and let $F_K$ denote the generating function of its appropriately normalized symmetrically colored HOMFLY-PT polynomials. A symmetric quiver is a quiver whose ad…

LMOV conjecture. The reformulated invariants have the displayed expansion with integer coefficients $\widetilde{\mathbf N}_{\mathbf R,i,j}$. This conjecture is presented as the LMO…

Let $\Xi_{\alpha}\equiv[\xi_1^{\alpha},\xi_2^{\alpha},\xi_3^{\alpha}]$ be an irreducible representation in $[r]^{\otimes3}$ having multiplicity $2$, and suppose … with the multipli…

The a-deformed $F_K$ conjecture. The function should also satisfy

Cyclotomic expansions. There exists such a function $C(K)$ for which

Kononov's figure-eight expansion. The $[r^s]$-colored HOMFLY-PT polynomial is expressed as

Let $K$ be a knot, let $P_r^K(a,q)$ denote its $r$-colored HOMFLY-PT polynomial, and let $\widehat{A}^K$ be the quantum $a$-deformed A-polynomial operator. A-polynomial recursion c…

Links–quivers correspondence conjecture. For each link $L$, there exists a quiver $Q$ such that the generating function $P_Q(\textbf{x})$ determines the generating function of the…

Let $B$ be a colored, oriented braid, and let $P(B)$ be the invariant of braid conjugacy classes with values in … where $\mathcal{I}$ is the ideal of relations between colored circ…

Let $\mathcal{L}$ be a link and let $p$ be a prime number. Write $\check{\mathcal{Z}}_p(\mathcal{L};q,t)$ for the reformulated colored HOMFLY-PT invariant associated with the $p$-r…