Random Knots in 3-Dimensional 3-Colour Percolation: Numerical Results and Conjectures

Abstract

Three-dimensional three-colour percolation on a lattice made of tetrahedra is a direct generalization of two-dimensional two-colour percolation on the triangular lattice. The interfaces between one-colour clusters are made of bicolour surfaces and tricolour non-intersecting and non-self-intersecting curves. Because of the three-dimensional space, these curves describe knots and links. The present paper presents a construction of such random knots using particular boundary conditions and a numerical study of some invariants of the knots. The results are sources of precise conjectures about the limit law of the Alexander polynomial of the random knots.

Random Colorings in Manifolds

“…We obtain new results in this setting, and settle an open problem. We show that the probability of an unknot occurring in this model decays at an exponential rate as the number of subdivisions increases, answering a question raised by de Crouy-Chanel and Simon [dCCS19]. We also rigorously establish other asymptotic properties, showing for example that the number of prime summands grows at least linearly.…”

“…In Section 5, we focus on the previously studied case d = k = 3, which produces random 1-dimensional submanifolds. This case leads to interesting percolation phenomena in an infinite 3-dimensional grid [SY14], and to models of random knots and links in a triangulated 3-sphere [dCCS19]. We obtain new results in this setting, and settle an open problem.…”

“…The following definition lists a selection of standard choices for the ambient triangulated manifold M , such that its vertices conveniently lie on an integer grid Z d . These constructions have been considered in previous works [SY14,dCCS19,BS20a], sometimes up to a linear transformation in R d , and sometimes equivalently in terms of the dual permutahedral tessellation and the coloring of its d-dimensional tiles. • The d-torus of order n, denoted T d n , is the quotient of the d-space R d /nZ d , or equivalently B d n with opposite (d − 1)-faces identified.…”

“…We now examine submanifolds obtained by randomly 3-coloring the interior vertices of the 3-cube B 3 n , which is the model of random knots introduced by de Crouy-Chanel and Simon [dCCS19]. The coloring of the boundary vertices is fixed using the following rule, which is sometimes called the beach ball coloring:…”

“…Sheffield and Yadin [SY14] used topological tools to analyze the emergence of large components in these random 1-manifolds in 3-space. Random knotting in these curves has been explored by de Crouy-Chanel and Simon [dCCS19], with numerical experiments and conjectures. These works inspired the general scheme developed here.…”