Vol. 6, No. 1, 2013

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Knots in the canonical book representation of complete graphs

Dana Rowland and Andrea Politano

Vol. 6 (2013), No. 1, 65–81
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 $n\le 11$.

Keywords
spatial graph, intrinsically knotted, canonical book
Mathematical Subject Classification 2010
Primary: 05C10, 57M15, 57M25