Geometry & Topology Monographs, Vol. 4 (2002),
Invariants of knots and 3-manifolds (Kyoto 2001),
Paper no. 5, pages 55--68.

On the quantum sl_2 invariants of knots and integral homology spheres

Kazuo Habiro

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.

Keywords. Quantum invariant, colored Jones polynomial, universal invariant, Witten-Reshetikhin-Turaev invariant

AMS subject classification. Primary: 57M27. Secondary: 17B37.

E-print: arXiv:math.GT/0211044

Submitted to GT on 30 November 2001. (Revised 8 April 2002.) Paper accepted 22 July 2002. Paper published 19 September 2002.

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Kazuo Habiro
Research Institute for Mathematical Sciences
Kyoto University, Kyoto, 606-8502, Japan

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