Analytic quantum invariants
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 Uq(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.
Related
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BibTeX
@misc{kashaev2026analytic,
title = {{Analytic quantum invariants}},
author = {Rinat Kashaev and Vladimir Mangazeev and Marcos Marino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24247},
eprint = {2610.24247},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24247},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Analytic quantum invariants
AU - Rinat Kashaev
AU - Vladimir Mangazeev
AU - Marcos Marino
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24247
UR - https://arxiv.org/abs/2610.24247
AB - 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(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.
ER -