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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)

Abstract

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

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

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

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