TY - JOUR

T1 - The expected total curvature of random polygons

AU - Cantarella, Jason

AU - Grosberg, Alexander Y.

AU - Kusner, Robert

AU - Shonkwiler, Clayton

N1 - Publisher Copyright:
©2015, 2015 Johns Hopkins University Press. All rights reserved.

PY - 2015

Y1 - 2015

N2 - 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.(formula presented) As a consequence, we show that at least 1/3 of fixed-length hexagons and 1/11 of fixed-length heptagons in ℝ3 are unknotted.

AB - 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.(formula presented) As a consequence, we show that at least 1/3 of fixed-length hexagons and 1/11 of fixed-length heptagons in ℝ3 are unknotted.

UR - http://www.scopus.com/inward/record.url?scp=84927593341&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84927593341&partnerID=8YFLogxK

U2 - 10.1353/ajm.2015.0015

DO - 10.1353/ajm.2015.0015

M3 - Article

AN - SCOPUS:84927593341

SN - 0002-9327

VL - 137

SP - 411

EP - 438

JO - American Journal of Mathematics

JF - American Journal of Mathematics

IS - 2

ER -