Volume 18, issue 2 (2018)

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Identifying lens spaces in polynomial time

Greg Kuperberg

Algebraic & Geometric Topology 18 (2018) 767–778

We show that if a closed, oriented 3–manifold M is promised to be homeomorphic to a lens space L(n,k) with n and k unknown, then we can compute both n and  k in polynomial time in the size of the triangulation of M. The tricky part is the parameter  k. The idea of the algorithm is to calculate Reidemeister torsion using numerical analysis over the complex numbers, rather than working directly in a cyclotomic field.

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3–manifolds, lens spaces, Reidemeister torsion
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 65G30, 68Q15, 68W01
Received: 26 April 2016
Revised: 31 December 2017
Accepted: 25 January 2018
Published: 12 March 2018
Greg Kuperberg
Department of Mathematics
University of California
Davis, CA
United States