Knots in the Canonical Book Representation of Complete Graphs

Authors

Dana Rowland, Merrimack CollegeFollow
Andrea Politano, Merrimack CollegeFollow

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+k_k) Hamiltonian cycles that are ambient isotopic to α 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 ≤ 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