Combinatorial duality for Poincaré series, polytopes and invariants of plumbed 3-manifolds

Abstract

Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the 'periodic constant' of the topological multivariable Poincaré series (zeta function). This involves a complicated regularization procedure (via quasipolynomials measuring the asymptotic behaviour of the coefficients).We show that the (a Gorenstein type) symmetry of the zeta function combined with Ehrhart-Macdonald-Stanley reciprocity (of Ehrhart theory of polytopes) provide a simple expression for the periodic cosntant. Using these dualities we also find a multivariable polynomial generalization of the Seiberg-Witten invariant, and we compute it in terms of lattice points of certain polytopes.All these invariants are also determined via lattice point counting, in this way we establish a completely general topological analogue of formulae of Khovanskii and Morales valid for singularities with non-degenerate Newton principal part.

On the topology of elliptic singularities

“…Remark 3.3.3.) The present proof shows the 'power' of the combination of [LNN17] and [LNN18]: they provide a surprisingly short proof for the p g -formula in this elliptic case.…”

“…We will write also M = M (Γ), where we think about it as the plumbed manifold associated with Γ. For more information and more details see [CDGZ04,CDGZ08,N11,NN02,BN10,LNN17,LNN18]. For an overview see also [N18,NBook].…”

“…In this note we try to bypass the theory of periodic constants and the theory of quasipolynomials associated with counting functions, since in the final arguments we will not need them; in this overview we mention them just to show the line of ideas behind the scenes. The point is that important surgery formulae are also formulated in terms of 'periodic constants' [BN10,LNN17,LNN18]. Here we will recall the most general (and recent) one.…”

The delta invariant of curves on rational surfaces II: Poincaré series and topological aspects

“…Finally, the twisted duality allows us to connect the periodic constant of P C with κ X (C), and to use different methods regarding the counting function of Z(t) (and developed recently by [23,24,21,22]) in order to find the following explicit formula for the δinvariant of a reduced curve germ (C, 0) embedded into a rational surface singularity (X, 0) as follows:…”

On the geometry of strongly flat semigroups and their generalizations

“…] (for the more general multivariable case see [LN14,LNN18]. In particular, P 0 determined all the integers {N (ℓ)} ℓ≥0 and p g as well.…”