Knots in the canonical book representation of complete graphs

Andrea Politano; Dana Rowland

doi:10.2140/involve.2013.6.65

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arXiv · 1106.4065

Knots in the canonical book representation of complete graphs

Andrea Politano·Dana Rowland

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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.

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Andrea Politano, Dana Rowland. 2011-06-21. Knots in the canonical book representation of complete graphs. https://doi.org/10.2140/involve.2013.6.65

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