Identifying lens spaces in polynomial time

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.