A Linear Time Algorithm for Visualizing Knotted Structures in 3 Pages

We introduce simple codes and fast visualization tools for knotted structures in molecules and neural 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 by an abstract Gauss code. First we recognize in linear time if an abstract Gauss code represents an actual 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 line. The running time of our drawing algorithm is linear in the length of a Gauss code of a given graph. Three-page embeddings provide simple linear codes of knotted graphs so that the isotopy problem for all graphs in 3-space completely reduces to a word problem in finitely presented semigroups.

[1]  Daniela Giorgi,et al.  Reeb graphs for shape analysis and applications , 2008, Theor. Comput. Sci..

[2]  Paul C. Kainen,et al.  The book thickness of a graph , 1979, J. Comb. Theory, Ser. B.

[3]  Hikoe Enomoto,et al.  Lower Bounds for the Number of Edge-crossings Over the Spine in a Topological Book Embedding of a Graph , 1999, Discret. Appl. Math..

[4]  Igor Muševič,et al.  Reconfigurable Knots and Links in Chiral Nematic Colloids , 2011, Science.

[5]  Pawel Dlotko,et al.  Computing fundamental groups from point clouds , 2015, Applicable Algebra in Engineering, Communication and Computing.

[6]  Vitaliy Kurlin,et al.  Gauss paragraphs of classical links and a characterization of virtual link groups , 2006, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  Mihalis Yannakakis,et al.  Embedding Planar Graphs in Four Pages , 1989, J. Comput. Syst. Sci..

[8]  Alexander J. Smola,et al.  Learning with kernels , 1998 .

[9]  L. Kauffman.  Invariants of graphs in three-space , 1989 .

[10]  V. Kurlin,et al.  Three-Page Embeddings of Singular Knots , 2003, math/0309325.

[11]  Emilio Di Giacomo,et al.  Curve-constrained drawings of planar graphs , 2005, Comput. Geom..

[12]  Mihalis Yannakais,et al.  Embedding planar graphs in four pages , 1989, STOC 1989.

[13]  Vitaliy Kurlin,et al.  THREE-PAGE ENCODING AND COMPLEXITY THEORY FOR SPATIAL GRAPHS , 2004 .