Knots in the canonical book representation of complete graphs

Details Knots in the canonical book representation of complete graphs Knots in the canonical book representation of complete graphs

2011-06-21

arxiv.org/abs/1106.4065v1

Mathematics

Geometric Topology

17 pages, 9 figures

Scientific paper

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.

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