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On the asymptotic expansion of the quantum SU(2) invariant at \(\zeta=e^{4\pi i/r}\) for Lens space \(L(p,q)\). (English) Zbl 1472.57022

The authors study the asymptotic expansion of the quantum \(SU(2)\) invariant for the lens space \(L(p,q)\) at \(\zeta=e^{\frac{4\pi i}{r}}\) with odd \(r\). The quantum \(SU(2)\) invariant at \(\zeta=e^{\frac{4\pi i}{r}}\) with odd \(r\) was defined by C. Blanchet et al. [Topology 34, No. 4, 883–927 (1995; Zbl 0887.57009)] following a skein-theoretic approach pioneered by Lickorish, extending the Reshetikhin-Turaev invariants associated to \(\zeta = e^{\frac{2\pi i}{r}}\). Asymptotic expansions for quantum \(SU(2)\) invariants at \(\zeta=e^{\frac{4\pi i}{r}}\) with odd \(r\) are closely related to \(SL(2;\mathbb{C})\) Chern-Simons theory, as it was recently observed by Q. Chen and T. Yang [Quantum Topol. 9, No. 3, 419–460 (2018; Zbl 1405.57020)] that the asymptotic expansion determines the hyperbolic volume of the 3-manifold. It was shown by T. Ohtsuki and T. Takata [Commun. Math. Phys. 370, No. 1, 151–204 (2019; Zbl 1441.57011)] that for some Seifert 3-manifolds those asymptotic expansions can be represented by a sum of contributions from \(SL(2;\mathbb{C})\) flat connections whose coefficients are square roots of the Reidemeister torsions.
The authors of this paper extend the analysis for lens spaces \(L(p,q)\), and confirm that the same is true for the lens spaces. The proof involves calculation of the quantum invariants for \(L(p,q)\) and explicit comparison of their asymptotic expansions with the square roots of Reidemeister torsions.

MSC:

57K31 Invariants of 3-manifolds (including skein modules, character varieties)
57K16 Finite-type and quantum invariants, topological quantum field theories (TQFT)
57K10 Knot theory
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