Vol. 13, No. 4, 2020

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Combinatorial random knots

Andrew Ducharme and Emily Peters

Vol. 13 (2020), No. 4, 633–654

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.

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knot theory, free knot diagrams, random knots, combinatorics
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57M27
Received: 24 February 2020
Revised: 30 June 2020
Accepted: 7 July 2020
Published: 20 November 2020

Communicated by Kenneth S. Berenhaut
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