Some conditionally hard problems on links and 3-manifolds

TL;DR

This paper demonstrates that several fundamental decision problems related to links and 3-manifolds are computationally hard, with some being NP-complete or at least as hard as graph isomorphism, under certain complexity conjectures.

Contribution

It establishes the computational hardness of three natural problems in topology, linking them to well-known complexity classes and conjectures.

Findings

Determining if a link bounds a Seifert surface with bounded Thurston norm is NP-complete.

Homeomorphism problem for closed 3-manifolds is at least as hard as graph isomorphism.

Deciding if a link is a sublink of another is NP-hard.

Abstract

We show that three natural decision problems about links and 3-manifolds are computationally hard, assuming some conjectures in complexity theory. The first problem is determining whether a link in the 3-sphere bounds a Seifert surface with Thurston norm at most a given integer; this is shown to be NP-complete. The second problem is the homeomorphism problem for closed 3-manifolds; this is shown to be at least as hard as the graph isomorphism problem. The third problem is determining whether a given link in the 3-sphere is a sublink of another given link; this is shown to be NP-hard.