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 .M; Z/ 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.

Calculating Heegaard–Floer homology by counting lattice points in tetrahedra

“…Theorem 1.14. A Brieskorn sphere Σ = Σ(p, q, r) is weakly elliptic if and only if (p, q, r) is equal to one of the following triplets: (3,4,5), (2,5,7), (2,5,9), or (2, 3, r) with gcd(6, r) = 1 and r > 5. There are no weakly elliptic Seifert homology spheres with more than three singular fibers.…”

“…Then l 3 since only Seifert homology with less than 3 singular fibers is S 3 . Suppose l = 3, then the triple (p 1 , p 2 , p 3 ) must be greater than or equal to one of the following triples: (3,5,7), (3,4,7), (3,5,9), (2,7,9), (2,5,11), (2,5,13), (2,5,19), (2,3,19). These triples are the immediate successors of the triples appearing in the table.…”

“…We would like to point out that the complexity mentioned above has already been studied as the normalized Seiberg-Witten invariant, or as the normalized Euler characteristic of Heegaard-Floer homology, in several places including [13], [11], [8], [7]. It is the topological candidate for geometric genus of certain analytical singularities whose links are rational homology spheres.…”

Némethi's division algorithm for zeta-functions of plumbed 3-manifolds

“…For the purpose to decode the Seiberg–Witten invariants of from (c.f. Section 2.2), [, Reduction theorem 5.4.2] has shown that the variables of can be reduced to the variables of the nodes of the graph. Therefore, we restrict our discussions to the reduced zeta‐functions defined by .…”

“…The multivariable polynomial part associated with (defined by [, Formula (32)], see also Formula ) is mainly a combination of the one‐ and two‐variable cases studied by [, ] corresponding to the structure of the orbifold graph . The vertices of are the nodes of and two of them are connected by an edge if the corresponding nodes in are connected by a path which consists only vertices with valency .…”