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

  • Marthe de Crouy-Chanel
  • Damien SimonEmail author


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.


Knot 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.


  1. 1.
    Alexander, J.W.: Topological invariants of knots and links. Trans. Am. Math. Soc. 30, 275–306 (1928). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bradley, R.M., Strenski, P.N., Debierre, J.-M.: A growing self-avoiding walk in three dimensions and its relation to percolation. Phys. Rev. A 45(12), 8513–8524 (1992). ADSCrossRefGoogle Scholar
  3. 3.
    Cha, J.C., Livingston, C.: Knotinfo: table of knot invariants, November 2018.
  4. 4.
    Diao, Y., Pippenger, N., Sumners, D.W.: On random knots. J. Knot Theory Ramif. 3(3), 419–429 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Even-Zohar, C.: Models of random knots. J. Appl. Comput. Topol. 1, 263–296 (2017). CrossRefzbMATHGoogle Scholar
  6. 6.
    Even-Zohar, C., Hass, J., Linial, N., Nowik, T.: Invariants of random knots and links. Discret. Comput. Geom. 56, 274–314 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gayet, D., Welschinger, J.-Y.: Expected topology of random real algebraic submanifolds. J. Inst. Math. Jussieu 14(4), 673–702 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Goldschmidt, C., Ueltschi, D., Windridge, P.: Quantum heisenberg models and their probabilisticrepresentations. Contemp. Math. 552, 177–224 (2011). CrossRefzbMATHGoogle Scholar
  9. 9.
    Kronheimer, P.B., Mrowka, T.S.: Khovanov homology is an unknot-detector. Publ. Math. IHES 113, 97–208 (2011). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Letendre, Thomas: Expected volume and euler characteristics of random submanifolds. J. Funct. Anal. 270(8), 3047–3110 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lickorish, W.B.R.: A introduction to knot theory. Graduate Texts in Mathematics. Springer, New York (1997). CrossRefzbMATHGoogle Scholar
  12. 12.
    Nahum, A., Chalker, J.T.: Universal statistics of vortex lines. Phys. Rev. E 85, 031141 (2012). ADSCrossRefGoogle Scholar
  13. 13.
    Nahum, A., Chalker, J.T., Serna, P., Ortu no, M., Somoza, A.M.: 3d loop models and the \({{\rm cp}}^{n-1}\) sigma model. Phys. Rev. Lett. 107, 110601 (2011). ADSCrossRefGoogle Scholar
  14. 14.
    Nahum, A., Chalker, J.T., Serna, P., Ortu no, M., Somoza, A.M.: Length distributions in loop soups. Phys. Rev. Lett. 111, 100601 (2013). ADSCrossRefGoogle Scholar
  15. 15.
    Sheffield, S., Yadin, A.: Tricolor percolation and random paths in 3d. Electron. J. Probab. 19(4), 1–23 (2014). MathSciNetzbMATHGoogle Scholar
  16. 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
  17. 17.
    Soteros, C.E., Sumners, D.W., Whittington, S.G.: Entanglement complexity of graphs in \({\mathbb{Z}}^3\). Math. Proc. Camb. Phil. Soc. 111, 75–91 (1992). CrossRefzbMATHGoogle Scholar
  18. 18.
    Taylor, A.J., Dennis, M.R.: Vortex knots in tangled quantum eigenfunctions. Nat. Commun. 7, 12346 (2016). ADSCrossRefGoogle Scholar
  19. 19.
    Wu, F.Y.: Knot theory and statistical mechanics. Rev. Mod. Phys. 64(4), 1099–1131 (1992). ADSMathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

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

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