Combinatorial Random Knots

TL;DR

This paper investigates the properties of free knot diagrams, analyzing their ability to produce various knots, especially trefoils, and explores probabilistic bounds on knot types arising from these diagrams.

Contribution

It introduces the study of free knot diagrams and their knot-producing capabilities, providing new insights and conjectures supported by computational data.

Findings

Every free knot diagram produces trefoil knots.

Certain simple free knot families are fully characterized.

Conjectures on probability bounds for unknot and trefoil outcomes.

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, or being the trefoil.