On the asymptotic expansion of the quantum SU(2) invariant at q = exp(4π∕N) for closed hyperbolic 3–manifolds obtained by integral surgery along the figure-eight knot

Ohtsuki, Tomotada¹

¹ Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto, Japan

Algebraic and Geometric Topology, Tome 18 (2018) no. 7, pp. 4187-4274

Résumé

It is known that the quantum SU(2) invariant of a closed 3–manifold at q = exp(2π−1∕N) is of polynomial order as N →∞. Recently, Chen and Yang conjectured that the quantum SU(2) invariant of a closed hyperbolic 3–manifold at q = exp(4π−1∕N) is of order exp(N ⋅ ς(M)), where ς(M) is a normalized complex volume of M. We can regard this conjecture as a kind of “volume conjecture”, which is an important topic from the viewpoint that it relates quantum topology and hyperbolic geometry.

In this paper, we give a concrete presentation of the asymptotic expansion of the quantum SU(2) invariant at q = exp(4π−1∕N) for closed hyperbolic 3–manifolds obtained from the 3–sphere by integral surgery along the figure-eight knot. In particular, the leading term of the expansion is exp(N ⋅ ς(M)), which gives a proof of the Chen–Yang conjecture for such 3–manifolds. Further, the semiclassical part of the expansion is a constant multiple of the square root of the Reidemeister torsion for such 3–manifolds. We expect that the higher-order coefficients of the expansion would be “new” invariants, which are related to “quantization” of the hyperbolic structure of a closed hyperbolic 3–manifold.

Détail

DOI : 10.2140/agt.2018.18.4187

Classification : 57M27, 57M50
Keywords: knot, $3$–manifold, quantum invariant, volume conjecture