Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

TL;DR

This paper proves that deciding the contractibility of compressed curves on 3-manifold boundaries is in NP, introduces polynomial-time algorithms for compressed curves on surfaces, and advances the understanding of normal subgroup membership in surface groups.

Contribution

It establishes the NP membership of the contractibility problem for compressed curves on 3-manifolds and provides new polynomial-time algorithms for compressed curves on surfaces.

Findings

Contractibility of compressed curves on surfaces can be decided in polynomial time.

The algorithm for 3-manifolds is fixed-parameter tractable in the manifold's complexity.

Polynomial-time solutions for normal subgroup membership problems in surface groups.

Abstract

In this paper we prove that the problem of deciding contractibility of an arbitrary closed curve on the boundary of a 3-manifold is in NP. We emphasize that the manifold and the curve are both inputs to the problem. Moreover, our algorithm also works if the curve is given as a compressed word. Previously, such an algorithm was known for simple (non-compressed) curves, and, in very limited cases, for curves with self-intersections. Furthermore, our algorithm is fixed-parameter tractable in the complexity of the input 3-manifold. As part of our proof, we obtain new polynomial-time algorithms for compressed curves on surfaces, which we believe are of independent interest. We provide a polynomial-time algorithm which, given an orientable surface and a compressed loop on the surface, computes a canonical form for the loop as a compressed word. In particular, contractibility of compressed curves on surfaces can be decided in polynomial time; prior published work considered only constant genus surfaces. More generally, we solve the following normal subgroup membership problem in polynomial time: given an arbitrary orientable surface, a compressed closed curve $\gamma$, and a collection of disjoint normal curves $\Delta$, there is a polynomial-time algorithm to decide if $\gamma$ lies in the normal subgroup generated by components of $\Delta$ in the fundamental group of the surface after attaching the curves to a basepoint.