Classification of Book Representations of $K_6$

Abstract

A book representation of a graph is a particular way of embedding a graph in three dimensional space so that the vertices lie on a circle and the edges are chords on disjoint topological disks. We describe a set of operations on book representations that preserves ambient isotopy, and apply these operations to $K_6$, the complete graph with six vertices. We prove there are exactly 59 distinct book representations for $K_6$, and we identify the number and type of knotted and linked cycles in each representation. We show that book representations of $K_6$ contain between one and seven links, and up to nine knotted cycles. Furthermore, all links and cycles in a book representation of $K_6$ have crossing number at most four.