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 SLnSL_n-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 $SL_n$-character variety of a surface $\mathfrak{S}$ equipped with an ideal triangulation $\lambda$. The first is the (stated) $SL_n$-skein algebra $\mathscr{S}(\mathfrak{S})$. The second $\overline{\mathcal{X}}(\mathfrak{S},\lambda)$ is the Fock and Goncharov's quantization of their $X$-moduli space. The quantum trace is an algebra homomorphism $\bar{tr}^X:\overline{\mathscr{S}}(\mathfrak{S})\to\overline{\mathcal{X}}(\mathfrak{S},\lambda)$ where the reduced skein algebra $\overline{\mathscr{S}}(\mathfrak{S})$ is a quotient of $\mathscr{S}(\mathfrak{S})$. When the quantum parameter is 1, the quantum trace $\bar{tr}^X$ coincides with the classical Fock-Goncharov homomorphism. This is a generalization of the Bonahon-Wong quantum trace map for the case $n=2$. We then define the extended Fock-Goncharov algebra $\mathcal{X}(\mathfrak{S},\lambda)$ and show that $\bar{tr}^X$ can be lifted to $tr^X:\mathscr{S}(\mathfrak{S})\to\mathcal{X}(\mathfrak{S},\lambda)$. We show that both $\bar{tr}^X$ and $tr^X$ are natural with respect to the change of triangulations. When each connected component of $\mathfrak{S}$ has non-empty boundary and no interior ideal point, we define a quantization of the Fock-Goncharov $A$-moduli space $\overline{\mathcal{A}}(\mathfrak{S},\lambda)$ and its extension $\mathcal{A}(\mathfrak{S},\lambda)$. We then show that there exist quantum traces $\bar{tr}^A:\overline{\mathscr{S}}(\mathfrak{S})\to\overline{\mathcal{A}}(\mathfrak{S},\lambda)$ and $tr^A:\mathscr{S}(\mathfrak{S})\hookrightarrow\mathcal{A}(\mathfrak{S},\lambda)$, where the second map is injective, while the first is injective at least when $\mathfrak{S}$ is a polygon. They are equivalent to the $X$-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 $n$th term is the $n$th colored Jones polynomial. The paper is concerned with the asymptotic behavior of the value of the $n$th colored Jones polynomial at e^{\a/n}, when \a is a fixed complex number and $n$ tends to infinity. We analyze this asymptotic behavior to all orders in $1/n$ when \a is a sufficiently small complex number. In addition, we give upper bounds for the coefficients and degree of the $n$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 $2 \pi i$. Our proofs use crucially the cyclotomic expansion of the colored Jones function, due to Habiro.Comment: 31 pages, 13 figure