Vol. 6, No. 1, 2013

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 4, 543–722
Issue 3, 363–541
Issue 2, 183–362
Issue 1, 1–182

Volume 16, 5 issues

Volume 15, 5 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 8 issues

Volume 11, 5 issues

Volume 10, 5 issues

Volume 9, 5 issues

Volume 8, 5 issues

Volume 7, 6 issues

Volume 6, 4 issues

Volume 5, 4 issues

Volume 4, 4 issues

Volume 3, 4 issues

Volume 2, 5 issues

Volume 1, 2 issues

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Editors' interests
 
Subscriptions
 
ISSN 1944-4184 (online)
ISSN 1944-4176 (print)
 
Author index
To appear
 
Other MSP journals
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