Drawings of Kn with the same rotation scheme are the same up to triangle-flips (Gioan’s Theorem)

Published:

Recommended citation: Arroyo, A., McQuillan, D., Richter, R.B. and Salazar, G., 2017. Drawings of Kn with the same rotation scheme are the same up to triangle-flips (Gioan’s Theorem). AUSTRALASIAN JOURNAL OF COMBINATORICS, 67(2), pp.131-144. https://arxiv.org/pdf/1512.09040.pdf

A good drawing of $$K_n$$ is a drawing of the complete graph with $$n$$ vertices in the sphere such that: no two edges with a common end cross; no two edges cross more than once; and no three edges all cross at the same point. Gioan's Theorem asserts that any two good drawings of Kn that have the same rotations of incident edges at every vertex are equivalent up to Reidemeister moves. At the time of preparation, 10 years had passed between the statement in the WG 2005 conference proceedings and our interest in the proposition. Shortly after we completed our preprint, Gioan independently completed a preprint.