4 problems

Let $L$ be a link with $|L|$ components, and let $P^{\bigwedge^j}_L(q,a)$ denote its $\bigwedge^j$-colored HOMFLY-PT polynomial. Define … Let $P_Q(\overline{x})$ be the generating…

For a sequence of positive integers $a_1,\ldots,a_{2m}$, let $M(a_1,\ldots,a_{2m})$ be the corresponding matrix and call a polynomial $f(q)=\sum_{i=0}^n a_iq^i$ unimodal if…

Asymptotic average genus conjecture.

Changing color on unknot components only shifts $q$-grading. There exist rational functions $F_j(L)\in \mathbb{Z}a^{\pm 1},s^{\pm 1},t^{\pm 1}$ such that, for all $i\geq j$,