Computational Complexity of Enumerative 3-Manifold Invariants
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Fix a finite group G. We analyze the computational complexity of the problem of counting homomorphisms π_1(X) → G, where X is a topological space treated as computational input. We are especially interested in requiring G to be a fixed, finite, nonabelian, simple group. We then consider two cases: when the input X=M is a closed, triangulated 3-manifold, and when X=S^3 ∖ K is the complement of a knot (presented as a diagram) in S^3. We prove complexity theoretic hardness results in both settings. When M is closed, we show that counting homomorphisms π_1(M) → G (up to automorphisms of G) is #P-complete via parsimonious Levin reduction---the strictest type of polynomial-time reduction. This remains true even if we require M to be an integer homology 3-sphere. We prove an analogous result in the case that X=S^3 ∖ K is the complement of a knot. Both proofs proceed by studying the action of the pointed mapping class group MCG_*(Σ) on the set of homomorphisms {π_1(Σ) → G} for an appropriate surface Σ. In the case where X=M is closed, we take Σ to be a closed surface with large genus. When X=S^3 ∖ K is a knot complement, we take Σ to be a disk with many punctures. Our constructions exhibit classical computational universality for a combinatorial topological quantum field theory associated to G. Our "topological classical computing" theorems are analogs of the famous results of Freedman, Larsen and Wang establishing the quantum universality of topological quantum computing with the Jones polynomial at a root of unity. Instead of using quantum circuits, we develop a circuit model for classical reversible computing that is equivariant with respect to a symmetry of the computational alphabet.
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