Inverted Habiro Series and its Residues

2025/09/26 by Svoboda, Josef
#FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2509.22610

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.

Citations

• The descendant colored Jones polynomials
• Inverted state sums, inverted Habiro series, and indefinite theta functions
• A unification of the ADO and colored Jones polynomials of a knot
• A two-variable series for knot complements
• BPS spectra and 3-manifold invariants
• Quantum Knot Invariants
• 3-Manifolds and 3d Indices
• The degree of a q-holonomic sequence is a quadratic quasi-polynomial
• The Jones slopes of a knot
• An integral form of the quantized enveloping algebra of sl₂ and its completions
• The colored Jones polynomials and the simplicial volume of a knot
• The Universal R-Matrix, Burau Representaion and the Melvin-Morton Expansion of the Colored Jones Polynomial
• The hyperbolic volume of knots from quantum dilogarithm

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