Identifying lens spaces in polynomial time

TL;DR

This paper presents a polynomial-time algorithm to identify lens spaces from 3-manifold triangulations by computing Reidemeister torsion numerically, effectively determining the parameters defining the lens space.

Contribution

It introduces a novel polynomial-time method for recognizing lens spaces and computing their parameters using numerical analysis of Reidemeister torsion.

Findings

Algorithm runs in polynomial time

Successfully computes lens space parameters n and k

Uses numerical analysis over complex numbers

Abstract

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.