The Hilbert Polynomial of Quandles and Colorings of Random Links

Abstract

Given a finite quandle $Q$, we study the average number of $Q$-colorings of the closure of a random braid in $B_n$ as $n$ varies. In particular we show that this number coincides with some polynomial $P_Q\in \mathbb{Q}[x]$ for $n\gg 0$. The degree of this polynomial is readily computed in terms of $Q$ as a quandle and these invariants are computed for all quandles with $|Q|\le 4$. Additionally we show that the methods in this paper allow to improve on the stability results of arXiv:0912.0325 from "periodic stability" to "stability".