On the Witten--Reshetikhin--Turaev invariants of torus bundles

TL;DR

This paper computes quantum invariants for certain torus bundle 3-manifolds, proving key conjectures about their asymptotic behavior and growth rates, especially for the SU(2) case with knots.

Contribution

It provides explicit computations and proofs of Witten's asymptotic expansion conjecture and the growth rate conjecture for torus bundles, extending to cases with knots.

Findings

Confirmed Witten's asymptotic expansion conjecture for these 3-manifolds.

Proved the growth rate conjecture in the SU(2) case, including manifolds with knots.

Extended results to trace -2 homeomorphisms, covering all torus bundles.

Abstract

By methods similar to those used by Lisa Jeffrey, we compute the quantum $\mathrm{SU}(N)$-invariants for mapping tori of trace $2$ homeomorphisms of a genus $1$ surface when $N = 2,3$ and discuss their asymptotics. In particular, we obtain directly a proof of a version of Witten's asymptotic expansion conjecture for these 3-manifolds. We further prove the growth rate conjecture for these 3-manifolds in the $\mathrm{SU}(2)$ case, where we also allow the 3-manifolds to contain certain knots. In this case we also discuss trace $-2$ homeomorphisms, obtaining -- in combination with Jeffrey's results -- a proof of the asymptotic expansion conjecture for all torus bundles.