The Computational Complexity of Knot Genus and Spanning Area

TL;DR

This paper proves that determining the genus of a knot and whether a curve bounds a surface of small area are computationally hard problems, specifically NP-complete or NP-hard, in 3-dimensional topology.

Contribution

It establishes the NP-completeness of the knot genus problem and NP-hardness of the minimal surface area problem in 3-manifolds, highlighting their computational difficulty.

Findings

Knot genus determination is NP-complete.

Deciding small-area bounding surfaces is NP-hard.

Results connect 3D topology problems with computational complexity theory.

Abstract

We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. Using similar ideas, we show that deciding whether a curve in a metrized PL 3-manifold bounds a surface of area less than a given constant C is NP-hard.