2024 Workshop on (Mostly) Matroids

Lawrence polytopes and some invariants of a graph

22 Aug 2024, 12:05

25m

S236 (IBS Science Culture Center)

Presentation (25 min)

Speaker

Changxin Ding

Description

We use two dual Lawrence polytopes $P$ and $P^{\ast }$ of a graph $G$ to study the graph. The $h$-vector of the graphic (resp. cographic) matroid complex associated to $G$ coincides with the $h^{\ast }$-vector of the Lawrence polytope $P$ (resp. $P^{\ast }$). In general, the $h$-vector is an invariant defined for an abstract simplicial complex, which encodes the number of faces of different dimensions. The $h^{\ast }$-vector, a.k.a. the $\delta$-polynomial, is an invariant defined for a rational polytope 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 the reduced divisors of the graph $G$ in a more geometric way.