Non-normal affine monoids, modules and Poincar\'e series of plumbed 3-manifolds

TL;DR

This paper introduces a non-normal affine monoid and modules linked to negative definite plumbed 3-manifolds, providing combinatorial formulas for topological invariants like Seiberg--Witten invariants.

Contribution

It constructs new algebraic structures associated with plumbed 3-manifolds and derives explicit combinatorial formulas for key topological invariants.

Findings

Constructed a non-normal affine monoid related to 3-manifolds.

Derived combinatorial formulas for Seiberg--Witten invariants.

Described the structure of the Poincaré series in terms of these algebraic objects.

Abstract

We construct a non-normal affine monoid together with its modules associated with a negative definite plumbed $3$-manifold $M$. In terms of their structure, we describe the $H_1(M,\mathbb{Z})$-equivariant parts of the topological Poincar\'e series. In particular, we give combinatorial formulas for the Seiberg--Witten invariants of $M$ and for polynomial generalizations defined in a previous paper of the authors.