All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials
In the 1990's, \textit{E. Witten} famously identified Chern-Simons theory on 3-manifolds as arising from topological string theory of the cotangent bundle in [Chern-Simons gauge theory as a string theory, in: \textit{H. Hofer} (ed.) et al., The Floer memorial volume. Basel: Birkhäuser. Prog. Math. 133, 637--678 (1995; Zbl 0844.58018)]. More recently in [\textit{V. Bouchard} et al., Commun. Math. Phys. 287, No. 1, 117--178 (2009; Zbl 1178.81214)], Bouchard, Klemm, Marino, and Pasquetti showed that amplitudes in topological string theory can be computed using the topological recursion of \textit{B. Eynard} and \textit{N. Orantin} defined in [Commun. Number Theory Phys. 1, No. 2, 347--452 (2007; Zbl 1161.14026)]. In light of this work, Dijkgraaf, Fuji, and Manabe conjectured that the topological recursion's wave function applied to the \(SL_2(\mathbb C)\) character variety of a knot complement should coincide with the colored Jones polynomial in [\textit{R. Dijkgraaf} et al., Nucl. Phys., B 849, No. 1, 166--211 (2011; Zbl 1215.81082)]. However their theory required the introduction of ad hoc constants in order to obtain a correct formula. Here, the authors propose an analogous approach using a non-perturbative wave function defined by \textit{B. Eynard} and \textit{M. Mariño} in [J. Geom. Phys. 61, No. 7, 1181--1202 (2011; Zbl 1215.81084)]. They conjecture that using this wave function applied to the \(SL_2(\mathbb C)\) character variety of a knot complement should coincide with the colored Jones polynomial without the introduction of constants. They verify this for the figure eight knot and for the once-punctured torus bundle \(L^2R\).
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