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

Tamás László and András Némethi

Geometry & Topology 18 (2014) 717–778

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 H1(M, ) 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.

$3$–manifolds, $\mathbb{Q}$–homology spheres, plumbed $3$–manifolds, Seiberg–Witten invariant, Ehrhart theory, equivariant Ehrhart polynomials, affine monoids, polytopes, periodic constant, surface singularities
Mathematical Subject Classification 2010
Primary: 14E15, 57M27
Secondary: 52B20, 06F05, 57R57
Received: 22 November 2012
Revised: 10 June 2013
Accepted: 19 July 2013
Published: 20 March 2014
Proposed: Ronald Stern
Seconded: Ronald Fintushel, Simon Donaldson
Tamás László
Central European University
Nador u. 9
1051 Budapest
András Némethi
MTA Rényi Institute of Mathematics
Reáltanoda u. 13-15
1053 Budapest