Ehrhart theory of polytopes and Seiberg–Witten invariants of plumbed $3$–manifolds

Abstract

Let $M$ be a rational homology sphere plumbed $3$–manifold associated with a connected negative-definite plumbing graph. We show that its Seiberg–Witten invariants equal certain coefficients of an equivariant multivariable Ehrhart polynomial. For this, we construct the corresponding polytope from the plumbing graph together with an action of $H_1\left(M,\mathbb{Z}\right)$ and we develop Ehrhart theory for them. At an intermediate level we define the ‘periodic constant’ of multivariable series and establish their properties. In this way, one identifies the Seiberg–Witten invariant of a plumbed $3$–manifold, the periodic constant of its ‘combinatorial zeta function’ and a coefficient of the associated Ehrhart polynomial. We make detailed presentations for graphs with at most two nodes. The two node case has surprising connections with the theory of affine monoids of rank two.