47 research outputs found
The Colored Jones Polynomial and the A-Polynomial of Knots
We study relationships between the colored Jones polynomial and the
A-polynomial of a knot. We establish for a large class of 2-bridge knots the AJ
conjecture (of Garoufalidis) that relates the colored Jones polynomial and the
A-polynomial. Along the way we also calculate the Kauffman bracket skein module
of all 2-bridge knots. Some properties of the colored Jones polynomial of
alternating knots are established.Comment: Typos and minor mistakes corrected. To appear in Advances in
Mathematic
Integrality of quantum 3-manifold invariants and rational surgery formula
We prove that the Witten-Reshetikhin-Turaev (WRT) SO(3) invariant of an
arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a
rational surgery formula for the unified invariant dominating WRT SO(3)
invariants of rational homology 3-spheres at roots of unity of order co-prime
with the torsion. As an application, we compute the unified invariant for
Seifert fibered spaces and for Dehn surgeries on twist knots. We show that this
invariant separates integral homology Seifert fibered spaces and can be used to
detect the unknot.Comment: 18 pages, Compositio Math. in pres