1. THE EXPECTED TOTAL CURVATURE OF RANDOM POLYGONS.
- Author
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CANTARELLA, JASON, GROSBERG, ALEXANDER Y., KUSNER, ROBERT, and SHONKWILER, CLAYTON
- Subjects
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CURVATURE measurements , *POLYGONS , *NUMERICAL analysis , *HEXAGONS , *ANGLES - 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 edge-length 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 π/2 n + π/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 R3 are unknotted. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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