Combinatorial random knots

Ducharme, Andrew; Peters, Emily

doi:10.2140/involve.2020.13.633

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Combinatorial random knots

Andrew Ducharme and Emily Peters

Vol. 13 (2020), No. 4, 633–654

DOI: 10.2140/involve.2020.13.633

Abstract

We explore free knot diagrams, which are projections of knots into the plane which don’t record over/under data at crossings. We consider the combinatorial question of which free knot diagrams give which knots and with what probability. Every free knot diagram is proven to produce trefoil knots, and certain simple families of free knots are completely worked out. We make some conjectures (supported by computer-generated data) about bounds on the probability of a knot arising from a fixed free diagram being the unknot, trefoil, or figure-eight knot.

Keywords

knot theory, free knot diagrams, random knots, combinatorics

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57M27

Milestones

Received: 24 February 2020

Revised: 30 June 2020

Accepted: 7 July 2020

Published: 20 November 2020

Communicated by Kenneth S. Berenhaut

Authors

Andrew Ducharme
Department of Mathematics and Statistics Loyola University Chicago Chicago, IL United States

Emily Peters
Department of Mathematics and Statistics Loyola University Chicago Chicago, IL United States

Vol. 13, No. 4, 2020