Georgia Institute of Technology College of Sciences

This is an ongoing project. We make use of two dual Lawrence polytopes $P$ and $P*$ of a graph $G$, to study invariants of the graph. The $h$-vector of the graphic (resp. cographic) matroid complex associated to $G$ coincides with the $h^*$-vector of the Lawrence polytope $P$ (resp. $P^*$). In general, the $h$-vector is an invariant defined for an abstract simplicial complex, which encodes the number of faces of different dimensions. The $h^*$-vector, a.k.a. the $\delta$-polynomial, is an invariant defined for a rational polytope, which is obtained by dilating the polytope. By dissecting the Lawrence polytopes, we may study the $h$-vectors associated to the graph $G$ at a finer level. In particular, we understand activities and reduced divisors of the graph $G$ in a more geometric way. I will try to make the talk self-contained.