Crossing numbers of random two-bridge knots

Abstract

In a previous work, the first and third authors studied a random knot model for all two-bridge knots using billiard table diagrams. Here we present a closed formula for the distribution of the crossing numbers of such random knots. We also show that the probability of any given knot appearing in this model decays to zero at an exponential rate as the length of the billiard table goes to infinity. This confirms a conjecture from the previous work. 1 knots are parametrized by the already-closed curve (cos(at + φ), cos(bt + θ), cos(ct + ψ)) where t ∈ [0, 2π], and φ, θ, ψ ∈ R are fixed phase shifts [BHJS94, JP98, HZ07, BDHZ09].Not all knots are harmonic or Lissajous. However, the diagrams resulting from the projection of Lissajous curves to the xy-plane give rise to all knots by suitable choice of the crossing information. This was shown by Lamm [Lam99] in the study of Fourier knots.Koseleff and Pecker [KP11b] prove a similar statement for the open harmonic path (cos at, cos bt), in their work on Chebyshev knots. They reparametrize it as (T a (t), T b (t)) using the Chebyshev polynomials T n (cos θ) = cos nθ, and show that any choice for the crossings can be realized by some z(t) = T c (t + φ). Vassiliev represents all knots by polynomials in [Vas90].

A lower bound on the average genus of a 2-bridge knot

“…Both [CK15] and [CEZK18] discuss moves on billiard table diagrams that are similar to Reidemeister moves: the internal reduction move that deletes a run + + + or − − − of three in a row and the external reduction move that deletes + + − or − − + only from the start of the word or − + + or + − − only from the end of the word.…”

“…The author with Krishnan [CK15] and with Even-Zohar and Krishnan [CEZK18] developed a random model for these a = 3 Chebyshev diagrams by taking a random string of {+, −} n , where n = b − 1 is the number of crossings of the diagram and where + and − correspond to the slope of the overstrand in the crossing: and , respectively. Because a = 3, the only knots appearing in this model are 2-bridge knots and the unknot.…”

The average genus of a 2-bridge knot is asymptotically linear

“…Baader, Kjuchukova, Lewark, Misev, and Ray [BKLMR19] previously showed that if c is sufficiently large, then c 4 ≤ g c . The proof of Theorem 1.1 uses the Chebyshev billiard table model for knot diagrams of Koseleff and Pecker [KP11b,KP11a] as presented by Cohen and Krishnan [CK15] and with Even-Zohar [CEZK18]. This model yields an explicit enumeration of the elements of K c as well as an alternating diagram in the format of Figure 2 for each element of K c .…”

“…A given 2-bridge knot has infinitely many descriptions as strings of various lengths in the symbols {+, −}. Cohen, Krishnan, and Evan-Zohar's work [CK15,CEZK18] lets us describe 2-bridge knots in this manner but with more control on the number of strings representing a given 2-bridge knot.…”