Dear Topologists and Friends of Topology, Elena will give the topology seminar tomorrow (Thursday).
Title: Oriented matroids and straight-edge embeddings of graphs
Abstract: Matroid theory is an abstract theory of dependence introduced by Whitney in 1935. It is a natural generalization of linear (in)dependence. Oriented matroids can be thought of as combinatorial abstractions of point configurations over the reals. To every linear (straight-edge) embedding of a graph one can associate an oriented matroid, and the oriented matroid captures enough information to determine which pairs of disjoint cycles in the embedded graph are linked. In this talk, we will introduce the basics of oriented matroids. Then we will discuss results about the type and number of links in linear embeddings of $K_9$, the complete graph on nine vertices, and in linear embeddings of the complete 3-partite graph $K_{3,3,1}$. This is joint work with Ramin Naimi.