Finding non-orientable surfaces in 3-manifolds

TL;DR

This paper proves that finding non-orientable surfaces of a given Euler genus in 3-manifolds is NP-hard, but solvable in NP when the genus is odd, advancing understanding of computational complexity in 3D topology.

Contribution

It establishes NP-hardness for the problem and provides an explicit algorithm for cases where the Euler genus is odd.

Findings

The problem is NP-hard in general.

The problem is in NP for odd Euler genus.

An explicit algorithm is provided for odd genus cases.

Abstract

We investigate the complexity of finding an embedded non-orientable surface of Euler genus $g$ in a triangulated $3$-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into $3$-manifolds. We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.