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.
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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 -