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The 3D-index of Dimofte-Gaiotto-Gukov is an interesting collection of$q$-series with integer coefficients parametrised by a pair of integers andassociated to a 3-manifold with torus boundary. In this note, we explain thestructure of the asymptotic expansions of the 3D-index when $q=e^{2\pi i\tau}$and $\tau$ tends to zero (to all orders and with exponentially small termsincluded), and discover two phenomena: (a) when $\tau$ tends to zero on a raynear the positive real axis, the horizontal asymptotics of the meromorphic3D-index match to all orders with the asymptotics of the Turaev-Viro invariantof a knot, in particular explaining the Volume Conjecture of Chen-Yang fromfirst principles, (b) when $\tau \to 0$ on the positive imaginary axis, thevertical asymptotics of the 3D-index involves periods of a plane curve (the$A$-polynomial), as opposed to algebraic numbers, explaining some predictionsof Hodgson-Kricker-Siejakowski and leading to conjectural identities betweenperiods of the $A$-polynomial of a knot and integrals of the Eulerbeta-function.

Periods, the meromorphic 3D-index and the Turaev--Viro invariant