Computing Invariants of Knotted Graphs Given by Sequences of Points in 3-Dimensional Space

Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


We design a fast algorithm for computing the fundamental group of the complement to any knotted polygonal graph in 3-space. A polygonal graph consists of straight segments and is given by sequences of vertices along edge-paths. This polygonal model is motivated by protein backbones described in the Protein Data Bank by 3D positions of atoms. Our KGG algorithm simplifies a knotted graph and computes a short presentation of the Knotted Graph Group containing powerful invariants for classifying graphs up to isotopy. We use only a reduced plane diagram without building a large complex representing the complement of a graph in 3-space.


Fundamental Group Protein Backbone Geometric Graph Isotopy Class Cubical Complex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was done during the EPSRC-funded secondment at Microsoft Research Cambridge, UK. We thank all reviewers for valuable comments and helpful suggestions.


  1. 1.
    Brendel, P., Dlotko, P., Ellis, G., Juda, M., Mrozek, M.: Computing fundamental groups from point clouds. Appl. Algebra Eng. Commun. Comput. 26, 27–48 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Crowell, R., Fox, R.: Introduction to Knot Theory. Graduate Texts in Mathematics, vol. 57. Springer, New York (1963)Google Scholar
  3. 3.
    Ellis, G.: HAP — Homological Algebra Programming package for GAP. Version 1.10.13 (2013). Available for download at
  4. 4.
    Fenn, R.: Tackling the trefoils. J. Knot Theory Ramif. 21, 1240004 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gordon, C., Luecke, J.: Knots are determined by their complements. J. Am. Math. Soc. 2, 371–415 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jamroz, M., Niemyska, W., Rawdon, E., Stasiak, A., Millett, K., Sulkowski, P., Sulkowska, J.: KnotProt: a database of proteins with knots and slipknots. Nucleic Acids Res. 1, 1–9 (2014)Google Scholar
  7. 7.
    Kauffman, L.: Invariants of graphs in three-space. Trans. AMS 311, 697–710 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Khatib, F., Weirauch, M., Rohl, C.: Rapid knot detection and application to protein structure prediction. Bioinformatics 14, 252–259 (2006)CrossRefGoogle Scholar
  9. 9.
    Koniaris K., Muthukumar, M.: Self-entanglement in ring polymers. J. Chem. Phys. 95, 2871–2881 (1991)CrossRefGoogle Scholar
  10. 10.
    Kurlin, V.: Three-page encoding and complexity theory for spatial graphs. J. Knot Theory Ramif. 16, 59–102 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kurlin, V.: Gauss paragraphs of classical links and a characterization of virtual link groups. Math. Proc. Camb. Philos. Soc. 145, 129–140 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kurlin, V.: A linear time algorithm for visualizing knotted structures in 3 pages. In: Proceedings of IVAPP: Information Visualization Theory and Applications, Berlin, pp. 5–16 (2015)Google Scholar
  13. 13.
    Kurlin, V., Lines, D.: Peripherally specified homomorphs of link groups. J. Knot Theory Ramif. 16, 719–740 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kurlin, V., Smithers, C.: A linear time algorithm for embedding arbitrary knotted graphs into a 3-page book. In: Computer Vision, Imaging and Computer Graphics Theory and Applications, pp. 99–122. Springer, Berlin (2016).
  15. 15.
    Letscher, D.: On persistent homotopy, knotting and the Alexander module. In: Proceedings of ITCS (2012)zbMATHGoogle Scholar
  16. 16.
    Millett, K.C., Rawdon. E.J., Stasiak, A.: Linear random knots and their scaling behaviour. Macromolecules 38, 601–606 (2005)CrossRefGoogle Scholar
  17. 17.
    Moriuchi, H.: An enumeration of theta-curves with up to 7 crossings. In: Proceedings of the East Asian School of Knots, Links and Related Topics, Seoul (2004)zbMATHGoogle Scholar
  18. 18.
    Taylor, W.: A deeply knotted protein structure and how it might fold. Nature 406, 916–919 (2000)CrossRefGoogle Scholar
  19. 19.
    Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math. (2) 87, 56–88 (1968)Google Scholar
  20. 20.
    Whitten, W.: Knot complements and groups. Topology 26, 41–44 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK

Personalised recommendations