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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