2022/09/06 by Stavros Garoufalidis, Garoufalidis, Stavros, Campbell Wheeler +1 · 3 citations
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2209.02843
openalex publication_date 2022/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The 3D-index of Dimofte-Gaiotto-Gukov is an interesting collection of q-series with integer coefficients parametrised by a pair of integers and associated to a 3-manifold with torus boundary. In this note, we explain the structure of the asymptotic expansions of the 3D-index when q=e2πiτ and τ tends to zero (to all orders and with exponentially small terms included), and discover two phenomena: (a) when τ tends to zero on a ray near the positive real axis, the horizontal asymptotics of the meromorphic 3D-index match to all orders with the asymptotics of the Turaev-Viro invariant of a knot, in particular explaining the Volume Conjecture of Chen-Yang from first principles, (b) when τ→ 0 on the positive imaginary axis, the vertical asymptotics of the 3D-index involves periods of a plane curve (the A-polynomial), as opposed to algebraic numbers, explaining some predictions of Hodgson-Kricker-Siejakowski and leading to conjectural identities between periods of the A-polynomial of a knot and integrals of the Euler beta-function.