A linear time algorithm for embedding arbitrary knotted graphs into a 3-page book

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. Threepage 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.