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 Kn, and we conjecture that the canonical book representation of Kn attains the least possible number of knotted cycles for any embedding of Kn. The canonical book representation of Kn 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 K2p+q. For each knotted Hamiltonian cycle α in the canonical book representation of Kn, there are at least 2kn+k k Hamiltonian cycles that are ambient isotopic to α in the canonical book representation of Kn+k. Finally, we list the number and type of all nontrivial knots that occur as cycles in the canonical book representation of Kn for n 11.

Keywords
spatial graph, intrinsically knotted, canonical book
Mathematical Subject Classification 2010
Primary: 05C10, 57M15, 57M25
Milestones
Received: 18 January 2012
Accepted: 2 August 2012
Published: 23 June 2013

Communicated by Joel Foisy
Authors
Dana Rowland
Department of Mathematics
Merrimack College
315 Turnpike Street
North Andover, MA 01845
United States
Andrea Politano
Merrimack College
315 Turnpike Street
North Andover, MA 01845
United States