On the quantum sl_2 invariants of knots and integral homology spheres

Abstract

We will announce some results on the values of quantum sl_2 invariants of knots and integral homology spheres. Lawrence's universal sl_2 invariant of knots takes values in a fairly small subalgebra of the center of the h-adic version of the quantized enveloping algebra of sl_2. This implies an integrality result on the colored Jones polynomials of a knot. We define an invariant of integral homology spheres with values in a completion of the Laurent polynomial ring of one variable over the integers which specializes at roots of unity to the Witten-Reshetikhin-Turaev invariants. The definition of our invariant provides a new definition of Witten-Reshetikhin-Turaev invariant of integral homology spheres.

arXiv commentPublished by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon4/paper5.abs.html

References in corpus (2)

• A Contribution of the Trivial Connection to Jones Polynomial and Witten's Invariant of 3d Manifolds I
• Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion

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