Monday, July 26, 3pm ET (8pm BST, 7am Tue NZST)
Matthew Sullivan, University of Waterloo
Rotation Systems and Simple Drawings of $K_n$

Abstract:

A complete rotation system on $n$ vertices is a collection consisting of cyclic permutations of the elements $[n]\backslash \{i\}$, one cyclic permutation for each $i$. If $D$ is a simple drawing of a labelled graph, then a rotation at vertex $v$ is the cyclic ordering of the edges at $v$. In particular, the collection of all vertex rotations of a simple drawing of $K_n$ is a rotation system.

If a complete rotation system can be represented by a simple drawing of $K_n$, then we call such a rotation system realizable. In 2011, Jan Kynčl published a proof using homotopy implying that if all 6 vertex rotation systems of an $n$ vertex rotation system $R_n$ are realizable, then $R_n$ is realizable.

In this talk, we will briefly review a full characterization of realizable rotation systems, present a structural characterization of edges and faces in simple drawings of $K_n$ and see a combinatorial proof of a weak characterization of realizable rotation systems.