Deciding contractibility of a non-simple curve on the boundary of a 3-manifold: A computational Loop Theorem

TL;DR

This paper introduces a polynomial space algorithm to determine if a boundary curve of a 3-manifold is contractible, improving previous bounds and providing an algorithmic version of the Loop Theorem.

Contribution

It presents the first specialized algorithm for contractibility of boundary curves in 3-manifolds, leveraging 3-manifold topology methods and an algorithmic Loop Theorem.

Findings

Algorithm runs in polynomial space and exponential time.

First algorithm specifically designed for boundary curve contractibility.

Provides an algorithmic Loop Theorem in polynomial space.

Abstract

We present an algorithm for the following problem. Given a triangulated 3-manifold M and a (possibly non-simple) closed curve on the boundary of M, decide whether this curve is contractible in M. Our algorithm runs in space polynomial in the size of the input, and (thus) in exponential time. This is the first algorithm that is specifically designed for this problem; it considerably improves upon the existing bounds implicit in the literature for the more general problem of contractibility of closed curves in a 3-manifold. The proof of the correctness of the algorithm relies on methods of 3-manifold topology and in particular on those used in the proof of the Loop Theorem. As a byproduct, we obtain an algorithmic version of the Loop Theorem that runs in polynomial space, and (thus) in exponential time.