Inverted Habiro Series and its Residues

Josef Svoboda

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arXiv · 2509.22610

Inverted Habiro Series and its Residues

Josef Svoboda

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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.

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Josef Svoboda. 2025-09-26. Inverted Habiro Series and its Residues. https://arxiv.org/abs/2509.22610

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