DeepAI
Log In Sign Up

Computational Complexity of Enumerative 3-Manifold Invariants

05/23/2018
by   Eric Samperton, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

07/12/2017

Computational complexity and 3-manifolds and zombies

We show the problem of counting homomorphisms from the fundamental group...
07/13/2019

Coloring invariants of knots and links are often intractable

Let G be a nonabelian, simple group with a nontrivial conjugacy class C ...
10/27/2020

Computation of Large Asymptotics of 3-Manifold Quantum Invariants

Quantum topological invariants have played an important role in computat...
05/28/2018

Quantum generalizations of the polynomial hierarchy with applications to QMA(2)

The polynomial-time hierarchy (PH) has proven to be a powerful tool for ...
01/18/2023

stateQIP = statePSPACE

Complexity theory traditionally studies the hardness of solving classica...
12/19/2019

On the hardness of finding normal surfaces

There are fundamental topological problems, such as unknot recognition a...
04/01/2022

Nondeterminism subject to output commitment in combinatorial filters

We study a class of filters – discrete finite-state transition systems e...