#### Document Type

Article - Open Access

#### Publication Title

Involved, a Journal of Mathematics

#### Publisher

Mathematical Science Publishers

#### Publication Date

6-2013

#### Abstract/ Summary

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*≥12. 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}. For each knotted Hamiltonian cycle in the canonical book representation of K

_{n}, there are at least 2

*(*

^{k}^{n+}

*) Hamiltonian cycles that are ambient isotopic to α in the canonical book representation of K*

^{k}_{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*≤ 11.

#### Repository Citation

Rowland, D.,
&
Politano, A.
(2013). Knots in the Canonical Book Representation of Complete Graphs. *Involved, a Journal of Mathematics, 6*(1), 65-81.

Available at: https://scholarworks.merrimack.edu/mth_facpub/7

#### Publisher Statement

©2013 Mathematical Sciences Publishers. *Involve, a Journal of Mathematics* website available from: msp.org/involve