Splice formulae for Poincaré series of integral homology spheres

Let M be a plumbed integral homology sphere 3-manifold associated with a connected negative definite plumbing graph $\mathrm{\Gamma }$. One of its most important invariants is its multivariable Poincaré series (or zeta function) $f_{\mathrm{\Gamma }}(\mathbf{t})$. Several numerical invariants can be read from $f_{\mathrm{\Gamma }}(\mathbf{t})$, or even from its polynomial part' $ \operatorname{Pol}_Γ(\mathbf{t})$. For example, the normalized’ Casson’s invariant equals ${\mathrm{Pol}}_{\mathrm{\Gamma }}(1)$. In this note we provide several splice (surgery) formulae for $f_{\mathrm{\Gamma }}$ and ${\mathrm{Pol}}_{\mathrm{\Gamma }}$ (reduced to the node variables, or to the node variables of connected sub-graphs). In this way, these global invariants can be recovered from a collection of certain smaller graphs. Recall that some (integral homology sphere) 3-manifold invariants are additive with respect to the splice decomposition (like the Casson’s invariant). However, some invariants need some `splice correction terms’. In our formulae the correction terms are easily computable one variable Alexander polynomials.