Random knots in 3-dimensional 3-colour percolation: numerical results and conjectures

M. de Crouy-Chanel, D. Simon,

Journal of Statistical Physics (2019) 176: 574 ,

https://arxiv.org/abs/1811.09066,

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.

Supplementary material: numerical data set for the paper.