Volume 18, issue 6 (2018)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 19
Issue 3, 1079–1618
Issue 2, 533–1078
Issue 1, 1–532

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
Author Index
To Appear
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Other MSP Journals
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
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.

Keywords
random knot, Petaluma, petal diagram, Delbruck–Frisch–Wasserman conjecture
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 60B05
References
Publication
Received: 14 January 2018
Revised: 18 May 2018
Accepted: 7 June 2018
Published: 18 October 2018
Authors
Chaim Even-Zohar
Mathematics Department
University of California at Davis
Davis, CA
United States
https://www.math.ucdavis.edu/~chaim
Joel Hass
Mathematics Department
University of California at Davis
Davis, CA
United States
https://www.math.ucdavis.edu/~hass
Nathan Linial
Department of Computer Science
The Hebrew University of Jerusalem
Jerusalem
Israel
http://www.cs.huji.ac.il/~nati
Tahl Nowik
Department of Mathematics
Bar-Ilan university
Ramat Gan
Israel
http://math.biu.ac.il/~tahl