Inverted Habiro Series and its Residues
Geometric Topology
2025-09-29 v1 Quantum Algebra
Abstract
We study the Gukov--Manolescu (GM) series of knots and the inverted Habiro series (IHS) proposed by S. Park. We give a new formula for IHS in terms of coefficients of the GM series and truncated theta functions. We prove a multiplication formula for IHS, constructing a natural ring, in analogy with work of Habiro. We study the residues of IHS and apply them to Dehn surgery formulas. We also give a curious relation between the asymptotics of the GM series at roots of unity and the Kashaev invariant.
Keywords
Cite
@article{arxiv.2509.22610,
title = {Inverted Habiro Series and its Residues},
author = {Josef Svoboda},
journal= {arXiv preprint arXiv:2509.22610},
year = {2025}
}
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