The Witten–Reshetikhin–Turaev invariant for links in finite order mapping tori I
Abstract. We state Asymptotic Expansion and Growth Rate conjectures for the Witten-Reshetikhin-Turaev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automorphisms of marked surfaces. Our approach is based upon geometric quantisation of the moduli space of parabolic bundles on the surface, which we show coincides with the construction of the Witten-Reshetikhin-Turaev invariants using conformal field theory, as was recently completed by Andersen and Ueno.
The homological content of the Jones representations at q = −1
Abstract. We generalize a discovery of Kasahara and show that the Jones representations of braid groups, when evaluated at q = −1, are related to the action on homology of a branched double cover of the underlying punctured disk. As an application, we prove for a large family of pseudo-Anosov mapping classes a conjecture put forward by Andersen, Masbaum, and Ueno by extending their original argument for the sphere with four marked points to our more general case.
On the Witten–Reshetikhin–Turaev invariants of torus bundles
Abstract. By methods similar to those used by Lisa Jeffrey, we compute the quantum 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 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.