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55
THE COLORED JONES POLYNOMIAL AND THE APOLYNOMIAL OF KNOTS
, 2006
"... 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 conje ..."
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Cited by 26 (3 self)
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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.
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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Cited by 24 (3 self)
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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.
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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Cited by 22 (1 self)
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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.
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 21 (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.
Nahm sums, stability and the colored jones polynomial
 Research in Mathematical Sciences
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The volume conjecture, perturbative knot invariants, and recursion relations for topological strings
 Nuclear Phys. B 849
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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 18 (6 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.
Knot state asymptotics I, AJ conjecture and abelian representations
"... 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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Cited by 11 (2 self)
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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.
Creative Telescoping for Holonomic Functions
"... Abstract The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of ..."
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Cited by 11 (5 self)
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Abstract The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of a survey article: the developments in this area during the last two decades are sketched and a selection of references is compiled in order to highlight the impact of creative telescoping in numerous contexts. 1
Quantum field theory and the volume conjecture, Interactions between hyperbolic geometry, quantum topology and number theory
 Contemp. Math
, 2011
"... The volume conjecture states that for a hyperbolic knot K in the threesphere S3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S3\K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a ..."
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Cited by 10 (3 self)
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The volume conjecture states that for a hyperbolic knot K in the threesphere S3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S3\K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a very nonobvious, nontrivial manner. The goal of the present lectures ∗ is to review the original statement of the volume conjecture and its recent extensions and generalizations, and to show how, in the most general context, the conjecture can be understood in terms of topological quantum field theory. In particular, we consider: a) generalization of the volume conjecture to families of incomplete hyperbolic metrics; b) generalization that involves not only the leading (volume) term, but the entire asymptotic expansion in 1/N; c) generalization to quantum group invariants for groups of higher rank; and d) generalization to arbitrary links in arbitrary threemanifolds. ∗These notes are based on lectures given by the authors at the workshops Interactions Between