Abstract. We prove that the N-colored Jones polynomial for the torus knotTs,t satisfies the second order difference equation, which reduces to the first order difference equation for a case ofT2,2m+1. We show that the A-polynomial of the torus knot can be derived from the difference equation. Also constructed is a q-hypergeometric type expression of the colored Jones polynomial forT2,2m+1. 1.

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Abstract: We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group GC, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative GC invariants turn out to be interesting topological invariants, which are very different from finite type (Vassiliev) invariants obtained in a theory with compact gauge group G. We explore various aspects of these invariants and present an example where we compute them explicitly to high loop order. We also introduce a notion of “arithmetic TQFT ” and conjecture (with supporting numerical evidence) that SL(2, C) Chern-Simons theory is an example of such a theory. CALT-68-2716 Contents

Consider the Chern-Simons topological quantum field theory with gauge group SU2 and level k. Given a knot in the 3-sphere, this theory associates to the knot exterior an element in a vector space. We call this vector the knot state and study its asymptotic properties when the level is large. The latter vector space being isomorphic to the geometric quantization of the SU2-character variety of the peripheral torus, the knot state may be viewed as a section defined over this character variety. We first conjecture that the knot state concentrates in the large level limit to the character variety of the knot. This statement may be viewed as a real and smooth version of the AJ conjecture. Our second conjecture says that the knot state in the neighborhood of abelian representations is a Lagrangian state. Using microlocal techniques, we prove these conjectures for the figure eight and torus knots. The proof is based on q-difference relations for the colored Jones polynomial. We also provide a new proof for the asymptotics of the Witten-Reshetikhin-Turaev invariant of the lens spaces and a derivation of the Melvin-Morton-Rozansky theorem from the two conjectures. 1

Abstract. We prove an ultrametric q-difference version of the Maillet-Malgrange theorem on the Gevrey nature of formal solutions of nonlinear analytic q-difference equations. Since degq and ordq define two valuations on C(q), we obtain, in particular, a result on the growth of the degree in q and the order at q of formal solutions of nonlinear q-difference equations, when q is a parameter. We illustrate the main theorem by considering two examples: a q-deformation of “Painlevé II”, for the nonlinear situation, and a q-difference equation satisfied by the colored Jones polynomials of the figure 8 knots, in the linear case. We also consider a q-analog of the Maillet-Malgrange theorem, both in the complex and in the ultrametric setting, under the assumption that |q | =1and a classical diophantine condition.

Abstract. We study q-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (non-commutative) A-polynomial of a knot. Using the method of guessing, we obtain this polynomial explicitly for the Kp = (−2, 3,3+2p) pretzel knots for p = −5,...,5. This is a particularly interesting family since the pairs (Kp, −K−p) are geometrically similar (in particular, scissors congruent) with similar character varieties. Our computation of the noncommutative A-polynomial (a) complements the computation of the A-polynomial of the pretzel knots done by the first author and Mattman, (b) supports the AJ Conjecture for knots with reducible A-polynomial and (c) numerically computes the Kashaev invariant of pretzel knots in linear time. In a later publication, we will use the numerical computation of the Kashaev invariant to numerically verify the Volume Conjecture for the above mentioned pretzel knots. 1. The colored Jones polynomial: a q-holonomic sequence of natural origin 1.1. Introduction. The colored Jones polynomial of a knot K in 3-space is a q-holonomic sequence of Laurent polynomials of natural origin in Quantum Topology [GL05]. As a canonical recursion relation for this sequence we choose the one with minimal order; this is the so-called non-commutative A-polynomial of a knot [Gar04]. Using the computational method of guessing with undetermined coefficients [Kau09a, Kau09b]

Abstract. We study relationships between the colored Jones polynomial and the A-polynomial of a knot. The AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the A-polynomial is established for a large class of two-bridge knots, including all twist knots. We formulate a weaker conjecture and prove that it holds for all two-bridge knots. Along the way we also calculate the Kauffman bracket skein module of the complements of two-bridge knots. Some properties of the colored Jones polynomial are established. The Jones polynomial was discovered by Jones in 1984 [Jo] and has made a revolution in knot theory. Despite many efforts little is known about the relationship between the Jones polynomial and classical topology invariants like the fundamental group. The A-polynomial of a knot, introduced in [CCGLS], describes more or less the representation space of the knot

Abstract The A-polynomial of a knot in S 3 defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL2C. Here, we show that a non-trivial knot in S 3 has a non-trivial A-polynomial. We deduce this from the gauge-theoretic work of Kronheimer and Mrowka on SU2-representations of Dehn surgeries on knots in S 3. As a corollary, we show that if a conjecture connecting the colored Jones polynomials to the A-polynomial holds, then the colored Jones polynomials distinguish the unknot. AMS Classification 57M25, 57M27; 57M50 Keywords Knot, A-polynomial, character variety, Jones polynomial