Geometry & Topology Monographs 4 (2002) 55–68

DOI: 10.2140/gtm.2002.4.55

Abstract

We will announce some results on the values of quantum sl₂ invariants of knots and integral homology spheres. Lawrence's universal sl₂ invariant of knots takes values in a fairly small subalgebra of the center of the h-adic version of the quantized enveloping algebra of sl₂. 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.

Keywords

quantum invariant, colored Jones polynomial, universal invariant, Witten-Reshetikhin-Turaev invariant

Mathematical Subject Classification

Primary: 57M27

Secondary: 17B37

References

Publication

Received: 30 November 2001
Revised: 8 April 2002
Accepted: 22 July 2002
Published: 19 September 2002

Authors