Knots in the canonical book representation of complete graphs

Abstract

We describe which knots can be obtained as cycles in the canonical book representation of the complete graph $K_n$, and we conjecture that the canonical book representation of $K_n$ attains the least possible number of knotted cycles for any embedding of $K_n$. The canonical book representation of $K_n$ contains a Hamiltonian cycle that is a composite knot if and only if $n\ge 12$. When $p$ and $q$ are relatively prime, the $\left(p,q\right)$ torus knot is a Hamiltonian cycle in the canonical book representation of $K_{2p+q}$. For each knotted Hamiltonian cycle $\alpha$ in the canonical book representation of $K_n$, there are at least $2^k\left(\genfrac{}{}{0.0pt}{}{n+k}{k}\right)$ Hamiltonian cycles that are ambient isotopic to $\alpha$ in the canonical book representation of $K_{n+k}$. Finally, we list the number and type of all nontrivial knots that occur as cycles in the canonical book representation of $K_n$ for $n\le 11$.