On the complexity of cusped non-hyperbolicity

TL;DR

This paper demonstrates that recognizing non-hyperbolic cusped 3-manifolds is in NP, assuming S^3-recognition is in coNP, by establishing NP membership for irreducible toroidal recognition and unconditionally for satellite knot recognition.

Contribution

It proves that certain 3-manifold recognition problems are in NP, including irreducible toroidal recognition, and unconditionally for satellite knot recognition, advancing computational topology.

Findings

Non-hyperbolic cusped 3-manifold recognition is in NP.

Irreducible toroidal recognition is in NP.

Satellite knot recognition is in NP unconditionally.

Abstract

We show that the problem of showing that a cusped 3-manifold M is not hyperbolic is in NP, assuming $S^3$-RECOGNITION is in coNP. To this end, we show that IRREDUCIBLE TOROIDAL RECOGNITION lies in NP. Along the way we unconditionally recover SATELLITE KNOT RECOGNITION lying in NP. This was previously known only assuming the Generalized Riemann Hypothesis. Our key contribution is to certify closed essential normal surfaces as essential in polynomial time in compact orientable irreducible $\partial$-irreducible triangulations. Our work is made possible by recent work of Lackenby showing several basic decision problems in 3-manifold topology are in NP or coNP.