The complexity of detecting taut angle structures on triangulations

TL;DR

This paper proves that detecting taut angle structures in 3-manifold triangulations is NP-complete but fixed-parameter tractable with respect to treewidth, impacting computational topology and related decision problems.

Contribution

It establishes the NP-completeness of detecting taut angle structures and introduces fixed-parameter tractability results based on treewidth, advancing understanding of computational complexity in 3-manifold topology.

Findings

Detection problem is NP-complete.

Fixed-parameter tractability in treewidth.

Implications for other 3-manifold decision problems.

Abstract

There are many fundamental algorithmic problems on triangulated 3-manifolds whose complexities are unknown. Here we study the problem of finding a taut angle structure on a 3-manifold triangulation, whose existence has implications for both the geometry and combinatorics of the triangulation. We prove that detecting taut angle structures is NP-complete, but also fixed-parameter tractable in the treewidth of the face pairing graph of the triangulation. These results have deeper implications: the core techniques can serve as a launching point for approaching decision problems such as unknot recognition and prime decomposition of 3-manifolds.