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The distribution of
knots in the Petaluma model
Chaim Even-Zohar, Joel Hass, Nathan Linial and Tahl Nowik |
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Algebraic & Geometric Topology 18 (2018) 3647–3667
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Abstract |
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The representation of knots by petal diagrams (Adams et al 2012) naturally defines a sequence of distributions on the set of knots. We establish some basic properties of this randomized knot model. We prove that in the random –petal model the probability of obtaining every specific knot type decays to zero as , the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the –petal model represents at least exponentially many distinct knots. Past approaches to showing, in some random models, that individual knot types occur with vanishing probability rely on the prevalence of localized connect summands as the complexity of the knot increases. However, this phenomenon is not clear in other models, including petal diagrams, random grid diagrams and uniform random polygons. Thus we provide a new approach to investigate this question. |
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Keywords
random knot, Petaluma, petal diagram,
Delbruck–Frisch–Wasserman conjecture
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Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 60B05
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References |
Publication
Received: 14 January 2018
Revised: 18 May 2018
Accepted: 7 June 2018
Published: 18 October 2018
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Authors |
| Chaim Even-Zohar | |
| Mathematics Department University of California at Davis Davis, CA United States |
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| https://www.math.ucdavis.edu/~chaim | |
| Joel Hass | |
| Mathematics Department University of California at Davis Davis, CA United States |
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| https://www.math.ucdavis.edu/~hass | |
| Nathan Linial | |
| Department of Computer Science The Hebrew University of Jerusalem Jerusalem Israel |
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| http://www.cs.huji.ac.il/~nati | |
| Tahl Nowik | |
| Department of Mathematics Bar-Ilan university Ramat Gan Israel |
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| http://math.biu.ac.il/~tahl | |
