Knots in the canonical book representation of complete graphs
Geometric Topology
2017-03-27 v1
Abstract
We describe which knots can be obtained as cycles in the canonical book representation of K_n, the complete graph on n vertices. We show that the canonical book representation of K_n contains a Hamiltonian cycle that is a composite knot if and only if n>11 and we show that 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}. Finally, we list the number and type of all non-trivial knots that occur as cycles in the canonical book representation of K_n for n<12. We conjecture that the canonical book representation of K_n attains the least possible number of knotted cycles for any embedding of K_n.
Cite
@article{arxiv.1106.4065,
title = {Knots in the canonical book representation of complete graphs},
author = {Andrea Politano and Dana Rowland},
journal= {arXiv preprint arXiv:1106.4065},
year = {2017}
}
Comments
17 pages, 9 figures