The efficient certification of knottedness and Thurston norm

TL;DR

This paper proves that key problems in 3-manifold topology, including knot triviality and Thurston norm computation, are in NP, enabling efficient certification of these properties.

Contribution

It establishes that determining knot triviality, Thurston norm, and incompressible boundary in 3-manifolds are in NP, providing new complexity bounds for these problems.

Findings

Knot triviality testing is in NP.

Thurston norm equality problem is in NP.

Incompressible boundary detection is in NP.

Abstract

We show that the problem of determining whether a knot in the 3-sphere is non-trivial lies in NP. This is a consequence of the following more general result. The problem of determining whether the Thurston norm of a second homology class in a compact orientable 3-manifold is equal to a given integer is in NP. As a corollary, the problem of determining the genus of a knot in the 3-sphere is in NP. We also show that the problem of determining whether a compact orientable 3-manifold has incompressible boundary is in NP.