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Models of random knots

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Abstract

The study of knots and links from a probabilistic viewpoint provides insight into the behavior of “typical” knots, and opens avenues for new constructions of knots and other topological objects with interesting properties. The knotting of random curves arises also in applications to the natural sciences, such as in the context of the structure of polymers. We present here several known and new randomized models of knots and links. We review the main known results on the knot distribution in each model. We discuss the nature of these models and the properties of the knots they produce. Of particular interest to us are finite type invariants of random knots, and the recently studied Petaluma model. We report on rigorous results and numerical experiments concerning the asymptotic distribution of such knot invariants. Our approach raises questions of universality and classification of the various random knot models.

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Acknowledgements

I wish to thank Joel Hass, Nati Linial, and Tahl Nowik for many hours of insightful discussions on knots, random knots and many other knotty subjects during our joint work. Their knowledge, vision and ideas are hopefully reflected in this article. All false conjectures are mine. I also wish to thank Robert Adler, Eric Babson, Rami Band, Itai Benjamini, Harrison Chapman, Moshe Cohen, Nathan Dunfield, Dima Jakobson, Sunder Ram Krishnan, Gal Lavi, Neal Madras, Igor Rivin, Dani Wise, and many others for valuable discussions on randomized knot models. Finally, I thank Moshe Cohen and the reviewers for their helpful suggestions. This article is based on the introduction to the author’s doctoral thesis at the Hebrew University of Jerusalem under the supervision of Nati Linial. It was supported by BSF Grant 2012188. This final version was completed while the author was a postdoctoral fellow at the Institute for Computational and Experimental Research in Mathematics (ICERM) during the Fall of 2016. The author was supported in part by the Rothschild fellowship.

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    Chaim Even-Zohar

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Even-Zohar, C. Models of random knots. J Appl. and Comput. Topology 1, 263–296 (2017). https://doi.org/10.1007/s41468-017-0007-8

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