Binary geometries and Chord diagrams

Chord diagrams are combinatorial objects that generalize triangulations of a polygon. They depend on two parameters: an integer n and a degree d less of equal to n. Given n and d the set of all chord diagams has the structure of a (polytopal) simplicial complex. The associated polytope include for d=1 the hypercube, d=n the associahedron and d=2 the pellytope. In this talk I will present the combinatorics of chord diagrams and how the induce a family of binary geometries. This is based on work in progress with Ömer Gürdogan (University of Southampton).