Volume conjectures for Reshetikhin-Turaev and Turaev-Viro invariants

Tian Yang (Texas A&M University)

Supported by numerical evidences, Chen and I conjectured that at the root of unity exp(2πi/r) instead of the usually considered root exp(πi/r), the Turaev-Viro and the Reshetikhin-Turaev invariants of a hyperbolic 3-manifold grow exponentially with growth rates respectively the hyperbolic and the complex volume of the manifold. This reveals a different asymptotic behavior of the relevant quantum invariants than that of Wittens invariants (that grow polynomially by the Asymptotic Expansion Conjecture), which may indicate a different geometric interpretation of those invariants than the SU(2) Chern-Simons gauge theory. In this talk, I will introduce the conjecture and show further supporting evidences, including recent joint works with Detcherry and Detcherry-Kalfagianni.