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2018 The distribution of knots in the Petaluma model
Chaim Even-Zohar, Joel Hass, Nathan Linial, Tahl Nowik
Algebr. Geom. Topol. 18(6): 3647-3667 (2018). DOI: 10.2140/agt.2018.18.3647

Abstract

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 n–petal model the probability of obtaining every specific knot type decays to zero as n, 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 n–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.

Citation

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Chaim Even-Zohar. Joel Hass. Nathan Linial. Tahl Nowik. "The distribution of knots in the Petaluma model." Algebr. Geom. Topol. 18 (6) 3647 - 3667, 2018. https://doi.org/10.2140/agt.2018.18.3647

Information

Received: 14 January 2018; Revised: 18 May 2018; Accepted: 7 June 2018; Published: 2018
First available in Project Euclid: 27 October 2018

zbMATH: 06990073
MathSciNet: MR3868230
Digital Object Identifier: 10.2140/agt.2018.18.3647

Subjects:
Primary: 57M25
Secondary: 60B05

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.18 • No. 6 • 2018
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