Random Knots in 3-Dimensional 3-Colour Percolation: Numerical Results and Conjectures
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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.
KeywordsKnot theory Percolation Numerical simulations Three-dimensional
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.
- 3.Cha, J.C., Livingston, C.: Knotinfo: table of knot invariants, November 2018. http://www.indiana.edu/~knotinfo/
- 16.Smirnov, S.: Towards conformal invariance of 2d lattice models. In: Sanz-Solé, M. (ed.) Proceedings of the International Congress of Mathematicians (ICM), 22–30 August 2006, vol. 2, pp. 1421–1451. European Mathematical Society, Madrid, Spain (2006)Google Scholar