Drawings of Kn with the same rotation scheme are the same up to Reidemeister moves. Gioan's Theorem

TL;DR

This paper proves that all good drawings of the complete graph with identical local rotations are equivalent under Reidemeister moves, confirming a conjecture related to graph drawing equivalences.

Contribution

The paper establishes Gioan's Theorem, showing that same-rotation good drawings of $K_n$ are equivalent via Reidemeister moves, resolving a longstanding conjecture.

Findings

Proof of Gioan's Theorem confirming equivalence of drawings

Clarification of conditions for drawing equivalence under Reidemeister moves

Resolution of a 10-year-old open problem in graph drawing theory

Abstract

A {\em 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 $K_n$ 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.