Random knots using Chebyshev billiard table diagrams

Abstract

Abstract. We use the Chebyshev knot diagram model of Koseleff and Pecker in order to introduce a random knot diagram model by assigning the crossings to be positive or negative uniformly at random. We give a formula for the probability of choosing a knot at random among all knots with bridge index at most 2. Restricted to this class, we define internal and external reduction moves that decrease the number of crossings of the diagram. We make calculations based on our formula, showing the numerics in graphs and providing evidence for our conjecture that the probability of any knot K appearing in this model decays to zero as the number of crossings goes to infinity.

Crossing numbers of random two-bridge knots

“…At this stage, we can derive precise and asymptotic expressions for P [K n = K], the probability that a given knot K of bridge number at most two occurs in the random model. The new closed form expression replaces the recursive solution derived in [CK15].…”

“…We first set up some notation and discuss background material on billiard table diagrams for two-bridge knots. We recall a few important definitions from the previous work [CK15], give some details on them as required here, and refer the reader to the relevant sections there for further clarifications.…”

“…This last setting was featured in the textbook "Random knotting and linking" edited by Millet and Sumners [MS94]. The reader can find more specific references in [EZHLN14] and [CK15]. Denote by T (a, b) the planar trajectory of a billiard ball in [0, a] × [0, b] fired at slope one from the lower left corner, where a and b are coprime integers.…”

“…Our analysis of the crossing number distribution relies on Theorem 4.1, where we derive a closed formula for the probability of obtaining any particular two-bridge knot in our random model. A recursive formula for this probability was given in Theorem 4.2 of [CK15].…”

Knots and Links from Random Projections

“…Over the past few decades numerous models of random knots and links have been studied and were developed either with a specific application in mind, or with the hope that the model will be sufficiently universal in the sense that it avoids biasing certain knots and links. Models include closed random walks on a lattice, random knots with confinement, random equilateral polygons, random kinematical links, random knots from billiard diagrams, random Fourier knots (for the aforementioned see [2], [4], [8], [11], for example), diagrams sampled from random 4-valent graphs (by Dunfield et. al...see SnapPy documentation: [9]), and the recent Petaluma model studied in [12].…”