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arXiv · 2610.24247

Analytic quantum invariants

Rinat Kashaev·Vladimir Mangazeev·Marcos Marino
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Abstract

Using a deformed Borel-Laplace resummation, we propose a framework for repackaging quantum invariants as analytic invariants. We illustrate our approach through a detailed analysis of the coloured Jones polynomials of several simple knots, deriving explicit expressions for the resulting analytic invariants in terms of $(q,X)$-series. These invariants are closely related to, but generally distinct from, those obtained via the conventional Borel resummation of the Melvin-Morton-Rozansky expansion. As a further example, we construct an analytic invariant from a sequence of two-variable polynomials conjectured to be the Links-Gould polynomials of the figure-eight knot, associated with the quantum superalgebras $U_q(\mathfrak{gl}(n|1))$. Finally, we apply our technique to $q$-series arising in the study of three-manifold invariants, revealing connections with mock modular forms.

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Rinat Kashaev, Vladimir Mangazeev, Marcos Marino. 2026-10-16. Analytic quantum invariants. https://arxiv.org/abs/2610.24247

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