The Expected Total Curvature of Random Polygons

TL;DR

This paper analyzes the expected total curvature of random polygons, providing formulas for various measures, and demonstrates that a significant fraction of fixed-length polygons are unknotted.

Contribution

It derives a formula for the expected total curvature of random polygons under natural measures and proves specific expectations for fixed-length polygons.

Findings

Expected total curvature approaches a constant difference for large polygons.

Exact formula for expected total curvature of fixed-length polygons in 3D.

At least 1/3 of hexagons and 1/11 of heptagons are unknotted.

Abstract

We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed polygon and a random open polygon with the same number of turning angles approaches a positive constant. We show that this is true for a natural class of probability measures on polygons, and give a formula for the constant in terms of the moments of the edgelength distribution. We then consider the symmetric measure on closed polygons of fixed total length constructed by Cantarella, Deguchi, and Shonkwiler. For this measure, we are able to prove that the expected value of total curvature for a closed n-gon is exactly \pi/2 n + (\pi/4) 2n/(2n-3). As a consequence, we show that at least 1/3 of fixed-length hexagons and 1/11 of fixed-length heptagons in 3-space are unknotted.