**Monday, July 26,** **3pm ET** (8pm BST, 7am Tue NZST)

**Matthew Sullivan,** University of Waterloo

**Rotation Systems and Simple Drawings of $K_n$**

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**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.

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##### 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.

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##### 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.