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

  • Marthe de Crouy-Chanel
  • Damien SimonEmail author
Article
  • 2 Downloads

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.

Keywords

Knot theory Percolation Numerical simulations Three-dimensional 

Notes

Acknowledgements

D. S. is partially funded by the Grant ANR-14CE25-0014 (ANR GRAAL). We thank Adam Nahum for comments on a first version of the present paper and a nice introduction to the existing results in the physics literature on similar models. We also thank the anonymous referees for additional references and very interesting suggestions.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Sorbonne Université, Laboratoire de probabilités, statistique et modélisation, UMR 8001ParisFrance

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