Inverted Habiro Series and its Residues

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.