A Linear Time Algorithm for Embedding Arbitrary Knotted Graphs into a 3-Page Book

  • Vitaliy KurlinEmail author
  • Christopher Smithers
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 598)


We introduce simple codes and fast visualization tools for knotted structures in complicated molecules and brain networks. Knots, links and more general knotted graphs are studied up to an ambient isotopy in Euclidean 3-space. A knotted graph can be represented by a plane diagram or a Gauss code. First we recognize in linear time if an abstract Gauss code represents a graph embedded in 3-space. Second we design a fast algorithm for drawing any knotted graph in the 3-page book, which is a union of 3 half-planes along their common boundary. The complexity of the algorithm is linear in the length of a Gauss code. Three-page embeddings are encoded in such a way that the isotopy classification for graphs in 3-space reduces to a word problem in finitely presented semigroups.

Supplementary material


  1. 1.
    Bernhart, F., Kainen, P.: The book thickness of a graph. J. Comb. Theory B 27, 320–331 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Biasotti, S., Giorgi, D., Spagnuolo, M., Falcidieno, B.: Reeb graphs for shape analysis and applications. Theoret. Comput. Sci. 392, 5–22 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boost C++ Libraries (version 1.59.0).
  4. 4.
    Brendel, P., Dlotko, P., Ellis, G., Juda, M., Mrozek, M.: Computing fundamental groups from point clouds. Appl. Algebra Eng. Commun. Comp. 26, 27–48 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.: Curve-constrained drawings of planar graphs. Comput. Geom. 30, 1–23 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Enomoto, H., Miyauchi, M.: Lower bounds for the number of edge-crossings over the spine in a topological book embedding of a graph. SIAM J. Discrete Math. 12, 337–341 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kauffman, L.: Invariants of graphs in three-space. Trans. AMS 311, 697–710 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kearton, C., Kurlin, V.: All 2-dimensional links live inside a universal 3-dimensional polyhedron. Algebraic Geom. Topology 8(3), 1223–1247 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kurlin, V.: Dynnikov three-page diagrams of spatial 3-valent graphs. Funct. Anal. Appl. 35(3), 230–233 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kurlin, V.: Three-page encoding and complexity theory for spatial graphs. J. Knot Theory Ramifications 16(1), 59–102 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kurlin, V.: Gauss paragraphs of classical links and a characterization of virtual link groups. Math. Proc. Cambridge Philos. Soc. 145(1), 129–140 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kurlin, V.: A linear time algorithm for visualizing knotted structures in 3 pages. In: Proceedings of Information Visualization Theory and Applications, IVAPP 2015, pp. 5–16 (2015)Google Scholar
  14. 14.
    Kurlin, V.: Computing invariants of knotted graphs given by sequences of points in 3-dimensional space. In: Proceedings of Topology-Based Methods in Visualization, TopoInVis 2015 (2015)Google Scholar
  15. 15.
    Kurlin, V., Vershinin, V.: Three-page embeddings of singular knots. Funct. Anal. Appl. 38(1), 14–27 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Schölkopf, B., Smola, A.: Learning with Kernels. MIT Press, Cambridge (2002)zbMATHGoogle Scholar
  17. 17.
    Smithers, C.: A linear time algorithm for embedding arbitrary knotted graphs into a 3-page book. MSc thesis, Durham University, UK (2015)Google Scholar
  18. 18.
    Tkalec, U., Ravnik, M., Copar, S., Zumer, S., Musevic, I.: Reconfigurable knots and links in chiral nematic colloids. Science 333(6038), 62–65 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Whitney, H.: Congruent graphs and the connectivity of graphs. Am. J. Math. 54(1), 150–168 (1932)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Yannakakis, M.: Embedding planar graphs in four pages. J. Comp. Syst. Sci. 38, 36–67 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Microsoft Research CambridgeCambridgeUK
  2. 2.Durham UniversityDurhamUK

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