Knots in the canonical book representation of complete graphs

TL;DR

This paper characterizes which knots appear as cycles in the canonical book representation of complete graphs, identifying specific conditions for complex knots and listing all non-trivial knots for small n.

Contribution

It provides a detailed analysis of knot types in the canonical book representation of K_n, including conditions for Hamiltonian cycles to be composite or torus knots, and lists all non-trivial knots for n<12.

Findings

Hamiltonian cycle is a composite knot if and only if n>11

(p,q) torus knot appears as a Hamiltonian cycle when p and q are relatively prime

Lists all non-trivial knots in K_n for n<12

Abstract

We describe which knots can be obtained as cycles in the canonical book representation of K_n, the complete graph on n vertices. We show that the canonical book representation of K_n contains a Hamiltonian cycle that is a composite knot if and only if n>11 and we show that when p and q are relatively prime, the (p,q) torus knot is a Hamiltonian cycle in the canonical book representation of K_{2p+q}. Finally, we list the number and type of all non-trivial knots that occur as cycles in the canonical book representation of K_n for n<12. We conjecture that the canonical book representation of K_n attains the least possible number of knotted cycles for any embedding of K_n.