Quantum invariants of random 3–manifolds

We consider the $SO (3)$ Witten–Reshetikhin–Turaev quantum invariants of random 3–manifolds. When the level $r$ is prime, we show that the asymptotic distribution of the absolute value of these invariants is given by a Rayleigh distribution which is independent of the choice of level. Hence the probability that the quantum invariant certifies the Heegaard genus of a random 3–manifold of a fixed Heegaard genus $g$ is positive but very small, less than $1 0^{- 7}$ except when $g \leq 3$ . We also examine random surface bundles over the circle and find the same distribution for quantum invariants there.

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Keywords

quantum invariants, random 3–manifolds, Heegaard genus

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 57N10

References

Publication

Received: 10 September 2010

Accepted: 21 June 2011

Published: 29 July 2011

Authors

Nathan M Dunfield
Department of Mathematics University of Illinois Urbana IL 61801 USA
http://dunfield.info/
Helen Wong
Department of Mathematics Carleton College 1 North College Street Northfield MN 55057 USA
http://people.carleton.edu/~hwong/

Volume 11, issue 4 (2011)

Nathan M Dunfield and Helen Wong

Abstract