#### Volume 18, issue 2 (2018)

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Identifying lens spaces in polynomial time

### Greg Kuperberg

Algebraic & Geometric Topology 18 (2018) 767–778
##### Abstract

We show that if a closed, oriented 3–manifold $M$ is promised to be homeomorphic to a lens space $L\left(n,k\right)$ 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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##### Keywords
3–manifolds, lens spaces, Reidemeister torsion
##### Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 65G30, 68Q15, 68W01