On the behavior of the colored Jones polynomial of the figure-eight knot under modular transformations
Christoph Aistleitner, Manuel Hauke
TL;DR
The paper resolves Zagier's continuity conjecture for the function $h(x)=\log\frac{\mathbf{J}_{4_1}(x)}{\mathbf{J}_{4_1}(1/x)}$ by proving that $h$ extends continuously to all irrationals, for the figure-eight knot $4_1$. It develops a detailed Ostrowski-number-based framework to decompose the Sudler products $P_N(x)$ into shifted factors, introduces a notion of good and evil indices via long runs of zeros in Ostrowski expansions, and analyzes the quotient of two quantum invariants via a running-factorization that becomes stable as the rational arguments converge to an irrational $\alpha$. The main novelty lies in handling the tail behavior without assuming unbounded partial quotients, using a running-index factorization and a robust control of perturbations, ultimately deducing a finite limit independent of the particular rational path to $\alpha$. This strengthens the bridge between quantum modular phenomena and Diophantine structure, with potential implications for broader classes of knots and $q$-series identities in quantum topology. The results thereby consolidate a sharp link between continued fractions, Sudler products, and the modular-type behavior predicted by Zagier.
Abstract
The colored Jones polynomial $J_{K,N}$ is an important quantum knot invariant in low-dimensional topology. In his seminal paper on quantum modular forms, Zagier predicted the behavior of $J_{K,0}(e^{2 πi x})$ under the action of $SL_2(\mathbb{Z})$ on $x \in \mathbb{Q}$. More precisely, Zagier made a prediction on the asymptotic value of the quotient $J_{K,0}(e^{2 πi γ(x)})/ J_{K,0}(e^{2 πi x})$ for fixed $γ\in SL_2(\mathbb{Z})$, as $x \to \infty$ along rationals with bounded denominator. In the case of the figure-eight knot $4_1$, which is the most accessible case, there is an explicit formula for $J_{4_1,0}(e^{2 πi x})$ as a sum of certain trigonometric products called Sudler products. By periodicity, the behavior of $J_{4_1,0}(e^{2 πi x})$ under the mapping $x \mapsto x+1$ is trivial. For the second generator of $SL_2(\mathbb{Z})$, Zagier conjectured that with respect to the mapping $x \mapsto 1/x$, the quotient $h(x) = \log ( J_{4_1,0}(e^{2 πi x}) / J_{4_1,0}(e^{2 πi /x}))$ can be extended to a function on $\mathbb{R}$ that is continuous at all irrationals. This conjecture was recently established by Aistleitner and Borda in the case of all irrationals that have an unbounded sequence of partial quotients in their continued fraction expansion. In the present paper we prove Zagier's continuity conjecture in full generality.