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The root polytope and the symmetric edge polytope are two types of lattice polytopes associated to graphs whose Ehrhart h^* polynomials have nice properties. For example, the h^*-polynomial of the root polytope is a generalization of T(x,1) where T(x,y) is the Tutte polynomial. I will talk about some results and conjectures related to these polytopes. Notably, we show how (somewhat unexpectedly) ribbon structures of the graph provide a useful tool to analyze these polytopes. Also, we formulate some conjectures on the facets of the symmetric edge polytope, that translate to questions on graph orientations.
Joint work with Tamás Kálmán. This talk is aimed at a general audience.