Topology Seminar (Main Talk): Coloring invariants of knots are often intractable

Seminar | November 29 | 4-5 p.m. | 3 Evans Hall

 Eric Samperton, University of California, Davis

 Department of Mathematics

Given a finite group $G$, we get an invariant of a knot $K$ by counting the number of $G$-colorings of $K$, i.e., the number of homomorphisms $\pi_1(S^3 \setminus K) \to G$. When $G$ is the dihedral group of order 6, this invariant is basically the number of Fox 3-colorings of $K$. We will show that when $G$ is non-abelian simple, the coloring invariant is #P-complete.

I discussed a similar theorem for closed 3-manifolds in this seminar back in January. In both cases, the argument involves an analysis of the mapping class group representations associated to a combinatorial, Aut$(G)$-equivariant TQFT. In this case, we have to understand Schur-type homological invariants of braid-equivalence classes of branched $G$-covers of the 2-sphere. After sketching the big picture of the proof, I’ll discuss these branched Schur invariants in more detail.