  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.

Geometric Topology;Quantum Algebra