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Difference equation of the colored Jones polynomial for torus knot
 Internat. J. Math
"... Abstract. We prove that the Ncolored 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 Apolynomial of the torus knot can be derived from the difference equation. Also c ..."
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Abstract. We prove that the Ncolored 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 Apolynomial of the torus knot can be derived from the difference equation. Also constructed is a qhypergeometric type expression of the colored Jones polynomial forT2,2m+1. 1.
Exact results for perturbative ChernSimons theory with complex gauge group
 Commun. Number Theory Phys
"... Abstract: We develop several methods that allow us to compute allloop partition functions in perturbative ChernSimons theory with complex gauge group GC, sometimes in multiple ways. In the background of a nonabelian irreducible flat connection, perturbative GC invariants turn out to be interestin ..."
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Abstract: We develop several methods that allow us to compute allloop partition functions in perturbative ChernSimons theory with complex gauge group GC, sometimes in multiple ways. In the background of a nonabelian 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) ChernSimons theory is an example of such a theory. CALT682716 Contents
Character varieties
"... Abstract. Let G be a complex reductive algebraic group and let Γ be a finitely generated group. In this paper we study properties of irreducible and completely reducible representations ρ: Γ → G in the context of the geometric invariant theory of the Gaction on Hom(Γ, G) by conjugation. In particul ..."
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Abstract. Let G be a complex reductive algebraic group and let Γ be a finitely generated group. In this paper we study properties of irreducible and completely reducible representations ρ: Γ → G in the context of the geometric invariant theory of the Gaction on Hom(Γ, G) by conjugation. In particular, we prove that ρ is polystable (i.e. its orbit is closed) if and only if ρ is completely reducible. We also show that ρ is properly stable (with respect to G/C(G)action) if and only if ρ is irreducible. The categorical quotient XG(Γ) = Hom(Γ, G)//G is the Gcharacter variety of Γ. We prove that if ρ is scheme smooth and completely reducible then T [ρ] XG(Γ) = T0(H 1 (Γ, Ad ρ)//SG(ρ)) where H 1 (Γ, Ad ρ) is the 1st cohomology group of Γ with coefficients in the lie algebra g of G twisted by the homomorphism Γ ρ
Knot state asymptotics I, AJ conjecture and abelian representations arXiv:1107.1645
"... Consider the ChernSimons topological quantum field theory with gauge group SU2 and level k. Given a knot in the 3sphere, 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 lat ..."
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Consider the ChernSimons topological quantum field theory with gauge group SU2 and level k. Given a knot in the 3sphere, 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 SU2character 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 qdifference relations for the colored Jones polynomial. We also provide a new proof for the asymptotics of the WittenReshetikhinTuraev invariant of the lens spaces and a derivation of the MelvinMortonRozansky theorem from the two conjectures. 1
AN ULTRAMETRIC VERSION OF THE MAILLETMALGRANGE THEOREM FOR NONLINEAR qDIFFERENCE EQUATIONS
"... Abstract. We prove an ultrametric qdifference version of the MailletMalgrange theorem on the Gevrey nature of formal solutions of nonlinear analytic qdifference 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 th ..."
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Abstract. We prove an ultrametric qdifference version of the MailletMalgrange theorem on the Gevrey nature of formal solutions of nonlinear analytic qdifference 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 qdifference equations, when q is a parameter. We illustrate the main theorem by considering two examples: a qdeformation of “Painlevé II”, for the nonlinear situation, and a qdifference equation satisfied by the colored Jones polynomials of the figure 8 knots, in the linear case. We also consider a qanalog of the MailletMalgrange theorem, both in the complex and in the ultrametric setting, under the assumption that q  =1and a classical diophantine condition.
THE NONCOMMUTATIVE APOLYNOMIAL OF (−2, 3, n) PRETZEL KNOTS
"... Abstract. We study qholonomic sequences that arise as the colored Jones polynomial of knots in 3space. The minimalorder recurrence for such a sequence is called the (noncommutative) Apolynomial of a knot. Using the method of guessing, we obtain this polynomial explicitly for the Kp = (−2, 3,3+2 ..."
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Abstract. We study qholonomic sequences that arise as the colored Jones polynomial of knots in 3space. The minimalorder recurrence for such a sequence is called the (noncommutative) Apolynomial 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 Apolynomial (a) complements the computation of the Apolynomial of the pretzel knots done by the first author and Mattman, (b) supports the AJ Conjecture for knots with reducible Apolynomial 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 qholonomic sequence of natural origin 1.1. Introduction. The colored Jones polynomial of a knot K in 3space is a qholonomic 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 socalled noncommutative Apolynomial of a knot [Gar04]. Using the computational method of guessing with undetermined coefficients [Kau09a, Kau09b]
THE COLORED JONES POLYNOMIAL AND THE APOLYNOMIAL OF KNOTS
, 2006
"... Abstract. We study relationships between the colored Jones polynomial and the Apolynomial of a knot. The AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the Apolynomial is established for a large class of twobridge knots, including all twist knots. We formulate a wea ..."
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Abstract. We study relationships between the colored Jones polynomial and the Apolynomial of a knot. The AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the Apolynomial is established for a large class of twobridge knots, including all twist knots. We formulate a weaker conjecture and prove that it holds for all twobridge knots. Along the way we also calculate the Kauffman bracket skein module of the complements of twobridge 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 Apolynomial of a knot, introduced in [CCGLS], describes more or less the representation space of the knot
ATG Nontriviality of the Apolynomial for knots in S 3
, 2004
"... Abstract The Apolynomial 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 nontrivial knot in S 3 has a nontrivial Apolynomial. We deduce this from the gaugetheoretic work of Kron ..."
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Abstract The Apolynomial 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 nontrivial knot in S 3 has a nontrivial Apolynomial. We deduce this from the gaugetheoretic work of Kronheimer and Mrowka on SU2representations of Dehn surgeries on knots in S 3. As a corollary, we show that if a conjecture connecting the colored Jones polynomials to the Apolynomial holds, then the colored Jones polynomials distinguish the unknot. AMS Classification 57M25, 57M27; 57M50 Keywords Knot, Apolynomial, character variety, Jones polynomial