709 research outputs found
Integrality of quantum 3-manifold invariants and a 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 Seifert fibered integral homology spaces and can be used to detect the unkno
Integrality of quantum 3-manifold invariants and a 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 Seifert fibered integral homology spaces and can be used to detect the unkno
Quantum traces for -skein algebras
We establish the existence of several quantum trace maps. The simplest one is
an algebra map between two quantizations of the algebra of regular functions on
the -character variety of a surface equipped with an ideal
triangulation . The first is the (stated) -skein algebra
. The second
is the Fock and Goncharov's
quantization of their -moduli space. The quantum trace is an algebra
homomorphism
where the reduced skein algebra is a
quotient of . When the quantum parameter is 1, the
quantum trace coincides with the classical Fock-Goncharov
homomorphism. This is a generalization of the Bonahon-Wong quantum trace map
for the case . We then define the extended Fock-Goncharov algebra
and show that can be lifted to
. We show
that both and are natural with respect to the change of
triangulations. When each connected component of has non-empty
boundary and no interior ideal point, we define a quantization of the
Fock-Goncharov -moduli space
and its extension . We then show that there
exist quantum traces
and
,
where the second map is injective, while the first is injective at least when
is a polygon. They are equivalent to the -versions but have
better algebraic properties.Comment: 111 pages, 35 figure
Asymptotics of the colored Jones function of a knot
To a knot in 3-space, one can associate a sequence of Laurent polynomials,
whose th term is the th colored Jones polynomial. The paper is concerned
with the asymptotic behavior of the value of the th colored Jones polynomial
at e^{\a/n}, when \a is a fixed complex number and tends to infinity.
We analyze this asymptotic behavior to all orders in when \a is a
sufficiently small complex number. In addition, we give upper bounds for the
coefficients and degree of the th colored Jones polynomial, with
applications to upper bounds in the Generalized Volume Conjecture. Work of
Agol-Dunfield-Storm-W.Thurston implies that our bounds are asymptotically
optimal. Moreover, we give results for the Generalized Volume Conjecture when
\a is near . Our proofs use crucially the cyclotomic expansion of
the colored Jones function, due to Habiro.Comment: 31 pages, 13 figure
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