The quantum A-polynomial conjecture for the knot series

Let KS3K\subset S^3 be a knot, let fK(x,q)f_K(x,q) be its rescaled knot series, let A^\widehat A be the quantum polynomial of KK, and let ΔK(x)\Delta_K(x) be its Alexander polynomial. Define the symmetric expansion s.e.\operatorname{s.e.} of a rational function in xx as the average of its Laurent expansion at x0x\to0 and its Laurent expansion in x1x^{-1} at xx\to\infty. Quantum A-polynomial conjecture. The quantum polynomial annihilates the knot series,

A^fK(x,q)=0,\widehat A f_K(x,q)=0,

and its classical limit is the symmetric expansion of the reciprocal Alexander polynomial,

limq1fK(x,q)=s.e.1ΔK(x).\lim_{q\to1}f_K(x,q)=\operatorname{s.e.}\frac{1}{\Delta_K(x)}.

This conjecture links the quantum polynomial recursion to the Alexander polynomial and is presented as a refinement motivated by the AJ conjecture; the source supplies no resolution status.

Sources & referencesView supporting material

Primary source

Miranda C. N. Cheng, Francesca Ferrari and Gabriele Sgroi, “Three-Manifold Quantum Invariants and Mock Theta Functions”, arXiv:1912.07997 (2020).

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