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

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