Volume 11, issue 4 (2011)

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Quantum invariants of random 3–manifolds

Nathan M Dunfield and Helen Wong

Algebraic & Geometric Topology 11 (2011) 2191–2205
Abstract

We consider the $SO\left(3\right)$ 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 $g\le 3$. We also examine random surface bundles over the circle and find the same distribution for quantum invariants there.

Keywords
quantum invariants, random 3–manifolds, Heegaard genus
Primary: 57M27
Secondary: 57N10
Publication
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/