Coloring invariants of knots and links are often intractable

Abstract

Let $G$ be a nonabelian, simple group with a nontrivial conjugacy class $C\subseteq G$. Let $K$ be a diagram of an oriented knot in $S^3$, thought of as computational input. We show that for each such $G$ and $C$, the problem of counting homomorphisms ${\pi }_1\left(S^3\setminus K\right)\to G$ that send meridians of $K$ to $C$ is almost parsimoniously $\#\mathsf{P}$–complete. This work is a sequel to a previous result by the authors that counting homomorphisms from fundamental groups of integer homology $3$–spheres to $G$ is almost parsimoniously $\#\mathsf{P}$–complete. Where we previously used mapping class groups actions on closed, unmarked surfaces, we now use braid group actions.