Georgia Institute of Technology College of Sciences

We will start by counting lattice points in a polytope and showhow this produces many familiar objects in mathematics.For example if one scales the polytope, the number of lattice points givesrise to the Ehrhart polynomials, including binomals and other well knownfunctions.Things get more interesting once we take a weighted sum over the latticepoints instead of just counting them. I will explain how toextend Ehrhart's theory in this case and discuss an application to knottheory. We will derive a new state sum for the colored HOMFLYpolynomial using q-Ehrhart polynomials, following my recent preprint Arxiv1501.00123.