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16
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.
Asymptotic behaviors of the colored Jones polynomial of a torus knot
, 2004
"... We study the asymptotic behaviors of the colored Jones polynomials of torus knots. Contrary to the works by R. Kashaev, O. Tirkkonen, Y. Yokota, and the author, they do not seem to give the volumes or the Chern–Simons invariants of the threemanifolds obtained by Dehn surgeries. On the other hand i ..."
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Cited by 10 (4 self)
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We study the asymptotic behaviors of the colored Jones polynomials of torus knots. Contrary to the works by R. Kashaev, O. Tirkkonen, Y. Yokota, and the author, they do not seem to give the volumes or the Chern–Simons invariants of the threemanifolds obtained by Dehn surgeries. On the other hand it is proved that in some cases the limits give the inverse of the Alexander polynomial.
GENERALIZED VOLUME CONJECTURE AND THE APOLYNOMIALS: THE NEUMANN–ZAGIER POTENTIAL FUNCTION AS A CLASSICAL LIMIT OF QUANTUM INVARIANT
, 2006
"... Abstract. We study quantum invariant Zγ(M) for cusped hyperbolic 3manifoldM. We construct this invariant based on oriented ideal triangulation ofMby assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in studies of the modular double of the quantum group. ..."
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Abstract. We study quantum invariant Zγ(M) for cusped hyperbolic 3manifoldM. We construct this invariant based on oriented ideal triangulation ofMby assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in studies of the modular double of the quantum group. Following Thurston and Neumann–Zagier, we deform a complete hyperbolic structure ofM, and correspondingly we define quantum invariant Zγ(Mu). This quantum invariant is shown to give the Neumann–Zagier potential function in the classical limitγ→0, and the Apolynomial can be derived from the potential function. We explain our construction by taking examples of 3manifolds such as complements of hyperbolic knots and punctured torus bundle over the circle. 1.
The sl3 jones polynomial of the trefoil: a case study of qholonomic recursions
"... Abstract. The sl3 colored Jones polynomial of the trefoil knot is a qholonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this qholonomic sequence as a case study. On the one hand, our result ..."
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Abstract. The sl3 colored Jones polynomial of the trefoil knot is a qholonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this qholonomic sequence as a case study. On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation concerning the colored Jones polynomial of a knot. On the other hand, this work illustrates the applicability and computational power of the employed computer algebra methods.
Guts of surfaces and the colored Jones polynomial
"... This work initiates a systematic study of relations between quantum and geometric knot invariants. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A – or B–adequacy), we derive direct and concrete relations between colored Jones polynomials and the ..."
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Cited by 2 (2 self)
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This work initiates a systematic study of relations between quantum and geometric knot invariants. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A – or B–adequacy), we derive direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. We prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement, and that certain coefficients of the polynomial measure how far this surface is from being a fiber in the knot complement. In particular, the surface is a fiber if and only if a particular coefficient vanishes. Our results also yield concrete relations between hyperbolic geometry and colored Jones polynomials: for certain families of links, coefficients of the polynomials determine the hyperbolic volume to within a factor of 4. Our methods here provide a deeper and more intrinsic explanation for similar connections that have been previously observed.
The Jones slopes of a knot
"... Abstract. The paper introduces an explicit Slope Conjecture that relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numb ..."
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Abstract. The paper introduces an explicit Slope Conjecture that relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural number) of a knot in 3space. We formulate a number of conjectures for these invariants and verify them by explicit computations for the class of alternating knots, the knots with at most 9 crossings, the torus knots and the (−2, 3, n) pretzel knots. Contents
On Habiro’s cyclotomic expansions of the Ohtsuki invariant
"... We give a selfcontained treatment of Le and Habiro’s approach to the Jones function of a knot and Habiro’s cyclotomic form of the Ohtsuki invariant for manifolds obtained by surgery around a knot. On the way we reproduce a state sum formula of Garoufalidis and Le for the colored Jones function of a ..."
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We give a selfcontained treatment of Le and Habiro’s approach to the Jones function of a knot and Habiro’s cyclotomic form of the Ohtsuki invariant for manifolds obtained by surgery around a knot. On the way we reproduce a state sum formula of Garoufalidis and Le for the colored Jones function of a knot. As a corollary, we obtain bounds on the growth of coefficients in the Ohtsuki series for manifolds obtained by surgery around a knot, which support the slope conjecture of Jacoby and the first author. 1
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
JONES POLYNOMIALS, VOLUME AND ESSENTIAL KNOT SURFACES: A SURVEY
, 2011
"... This paper is a brief overview of recent results by the authors relating colored Jones polynomials to geometric topology. The proofs of these results appear in the papers [14, 19], while this survey focuses on the main ideas and examples. ..."
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This paper is a brief overview of recent results by the authors relating colored Jones polynomials to geometric topology. The proofs of these results appear in the papers [14, 19], while this survey focuses on the main ideas and examples.