The expected total curvature of random polygons

Jason Cantarella, Alexander Y. Grosberg, Robert Kusner, Clayton Shonkwiler

    Research output: Contribution to journalArticlepeer-review


    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.

    Original languageEnglish (US)
    Pages (from-to)411-438
    Number of pages28
    JournalAmerican Journal of Mathematics
    Issue number2
    StatePublished - 2015

    ASJC Scopus subject areas

    • General Mathematics


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