Skip to main content
Unpublished Paper
The Expected Total Curvature of Random Polygons
American Journal of Mathematics (2015)
  • Jason Cantarella
  • Alexander Y. Grosberg
  • Robert Kusner, University of Massachusetts - Amherst
  • Clayton Shonkwiler

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.

Publication Date
April, 2015
Pre-published version downloaded from archive Published version located at
Citation Information
Jason Cantarella, Alexander Y. Grosberg, Robert Kusner and Clayton Shonkwiler. "The Expected Total Curvature of Random Polygons" American Journal of Mathematics (2015)
Available at: