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arXiv · 1206.2552

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

Jørgen Ellegaard Andersen·Søren Fuglede Jørgensen
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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.

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Jørgen Ellegaard Andersen, Søren Fuglede Jørgensen. 2014-05-06. On the Witten--Reshetikhin--Turaev invariants of torus bundles. https://arxiv.org/abs/1206.2552

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