The Knotting of Equilateral Polygons in R³

Abstract

It was proved in [4] that the knotting probability of a Gaussian random polygon goes to 1 as the length of the polygon goes to infinity. In this paper, we prove the same result for the equilateral random polygons in R3. More precisely, if EPn is an equilateral random polygon of n steps, then we have [Formula: see text] provided that n is large enough, where ∊ is some positive constant.

Models of random knots

“…Diao et al (1994) proved expðÀn e Þ for some e [ 0 for Gaussian-steps polygons. This was extended to equilateral polygons (Diao 1995) and other models (Janse et al 2007). …”

“…This is based on random samples of 25 knots with 21 to 121 petals. Non-hyperbolic knots (\ 2.5%) were omitted models (Sumners and Whittington 1988;Pippenger 1989;Diao et al 1994;Diao 1995). We do know, however, that large scale knotting occurs as well (Jungreis 1994;Diao et al 2001).…”

Knots, Slipknots, and Ephemeral Knots in Random Walks and Equilateral Polygons

“…With increasing length, the probability that a random walk or polygon contains a knot goes to one [SW88,Pip89,DPS94,Dia95] proving a conjecture of Frisch and Wasserman [FW61] and of Delbruck [Del62]. In addition, this knotted portion can be of any desired topological type.…”

“…The model for the proof is the following theorem of Yuanan Diao [Dia95], itself a generalization of the Gaussian random knot proof [DPS94]:…”

“…(Diao [Dia95]) All except exponentially few sufficiently long self-avoiding random walks and polygons in 3-space contain knots of any fixed topological type.…”