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Coloring invariants of knots and links are often intractable

Greg Kuperberg and Eric Samperton

Algebraic & Geometric Topology 21 (2021) 1479–1510

Let G be a nonabelian, simple group with a nontrivial conjugacy class C G. Let K be a diagram of an oriented knot in S3, thought of as computational input. We show that for each such G and C, the problem of counting homomorphisms π1(S3 K) G that send meridians of K to C is almost parsimoniously #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 #P–complete. Where we previously used mapping class groups actions on closed, unmarked surfaces, we now use braid group actions.

NP–hardness, \#P–hardness, knot invariants
Mathematical Subject Classification 2010
Primary: 20F10, 57M27, 68Q17
Received: 20 November 2019
Revised: 6 September 2020
Accepted: 22 September 2020
Published: 11 August 2021
Greg Kuperberg
Department of Mathematics
University of California, Davis
Davis, CA
United States
Eric Samperton
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL
United States