# Classification of Book Representations of $K_6$

## TL;DR

This paper classifies all 59 distinct book representations of the complete graph K_6 in three-dimensional space, analyzing their knotted and linked cycles, and establishing bounds on their complexity.

## Contribution

It provides a complete enumeration and classification of book representations of K_6, including the number and types of knotted and linked cycles, with bounds on crossing numbers.

## Key findings

- Exactly 59 distinct book representations of K_6 identified
- Book representations contain between one and seven links and up to nine knotted cycles
- All links and cycles have crossing number at most four

## 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.