#### 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.

Notes on file formats
Kazuo Habiro

Research Institute for Mathematical Sciences

Kyoto University, Kyoto, 606-8502, Japan

Email: habiro@kurims.kyoto-u.ac.jp

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