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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 ascomputational input. We show that for each such $G$ and $C$, the problem ofcounting homomorphisms $\pi_1(S^3\setminus K) \to G$ that send meridians of $K$to $C$ is almost parsimoniously $\mathsf{\#P}$-complete. This work is a sequelto a previous result by the authors that counting homomorphisms fromfundamental groups of integer homology 3-spheres to $G$ is almostparsimoniously $\mathsf{\#P}$-complete. Where we previously used mapping classgroups actions on closed, unmarked surfaces, we now use braid group actions.

Coloring invariants of knots and links are often intractable