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

Abstract

A polynomial counterpart of the Seiberg–Witten invariant associated with a negative definite plumbing 3‐manifold has been proposed by earlier work of the authors. It is provided by a special decomposition of the zeta‐function defined by the combinatorics of the manifold. In this article we give an algorithm, based on multivariable Euclidean division of the zeta‐function, for the explicit calculation of the polynomial, in particular for the Seiberg–Witten invariant.

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

“…Though this connection suggested the existence of a 'polynomial -negative degree part' decomposition of multivariable topological Poincaré series, hence a polynomial generalization of the Seiberg-Witten invariant, and it generated an intense activity (see e.g. [LN14,LSz16,LSz17]), the complete explanation waited till the present manuscript.…”

“…It satisfied (c), but it didn't answer (a) in a natural way. Later, [LSz17] considered another polynomial P + 2 (with P + 2 = P + 1 in general) constructed via an inductive multivariable Euclidean division. It answered (a)-(b), but (c) was not established, so it was not clear if P + 2 is helpful at all in pc (or Seiberg-Witten) computations.…”

“…We review in short the needed statement of [LSz17] reorganized from the point of view of the present note. We start, similarly as in section 3, with a pair of free Z-modules L ⊂ L ′ , and a general multivariable rational function (in variables t L ′ )…”