New York Journal of Mathematics
Volume 13 (2007) 11-16


Paul Blain, Garry Bowlin, Joel Foisy, Jacob Hendricks, and Jason LaCombe

Knotted Hamiltonian cycles in spatial embeddings of complete graphs

Published: February 20, 2007
Keywords: Spatial graph, embedded graph, intrinsically knotted
Subject: 57M15, 57M25

We show the complete graph on n vertices contains a knotted Hamiltonian cycle in every spatial embedding, for n > 7. Moreover, we show that for n >8, the minimum number of knotted Hamiltonian cycles in every embedding of Kn is at least (n-1)(n-2)...(9)(8). We also prove K8 contains at least 3 knotted Hamiltonian cycles in every spatial embedding.


The results in Section 3 were obtained during an NSF and NSA-sponsored summer Research Experience for Undergraduates.

Author information

Paul Blain:
Department of Mathematics, University of Washington, Seattle, WA 98195-4350

Garry Bowlin:
Department of Mathematics, Binghamton University, Binghamton, NY 13902

Joel Foisy:
Department of Mathematics, SUNY Potsdam, Potsdam, NY 13676

Jacob Hendricks:
Department of Mathematics, University of Texas at Austin, Austin, TX 78712

Jason LaCombe:
Department of Biostatistics and Computational Biology, University of Rochester, Rochester, NY 14642