Splice formulae for Poincaré series of integral homology spheres
Abstract
Let M be a plumbed integral homology sphere 3-manifold associated with a connected negative definite plumbing graph Γ. One of its most important invariants is its multivariable Poincaré series (or zeta function) fΓ(t). Several numerical invariants can be read from fΓ(t), or even from its `polynomial part' $ \operatorname{Pol}_Γ(\mathbf{t})$. For example, the `normalized' Casson's invariant equals $ \operatorname{Pol}_Γ(1)$. In this note we provide several splice (surgery) formulae for fΓ and PolΓ (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.
Related
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BibTeX
@misc{laszlo2026splice,
title = {{Splice formulae for Poincaré series of integral homology spheres}},
author = {Tamás László and András Némethi and György Tőtős},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16840},
eprint = {2609.16840},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16840},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Splice formulae for Poincaré series of integral homology spheres
AU - Tamás László
AU - András Némethi
AU - György Tőtős
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16840
UR - https://arxiv.org/abs/2609.16840
AB - Let M be a plumbed integral homology sphere 3-manifold associated with a connected negative definite plumbing graph Γ. One of its most important invariants is its multivariable Poincaré series (or zeta function) f_Γ(t). Several numerical invariants can be read from f_Γ(t), or even from its `polynomial part' $ \operatorname{Pol}_Γ(\mathbf{t})$. For example, the `normalized' Casson's invariant equals $ \operatorname{Pol}_Γ(1)$. In this note we provide several splice (surgery) formulae for f_Γ and Pol_Γ (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.
ER -