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The distribution of knots in the Petaluma model

Chaim Even-Zohar, Joel Hass, Nathan Linial and Tahl Nowik

Algebraic & Geometric Topology 18 (2018) 3647–3667

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.

random knot, Petaluma, petal diagram, Delbruck–Frisch–Wasserman conjecture
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 60B05
Received: 14 January 2018
Revised: 18 May 2018
Accepted: 7 June 2018
Published: 18 October 2018
Chaim Even-Zohar
Mathematics Department
University of California at Davis
Davis, CA
United States
Joel Hass
Mathematics Department
University of California at Davis
Davis, CA
United States
Nathan Linial
Department of Computer Science
The Hebrew University of Jerusalem
Tahl Nowik
Department of Mathematics
Bar-Ilan university
Ramat Gan