The coloured Jones function

Melvin, Paul; Morton, H. R.

doi:10.1007/bf02099310

Cited by 91 publications

(96 citation statements)

References 5 publications

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“…(This result generalizes a conjecture of Melvin and Morton [61], proved in [2].) More precisely, Rozansky proved that J MMR K has the following power series expansion…”

Section: Remarks On Knot Invariantssupporting

confidence: 80%

A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres

Habiro

2007

Invent. math.

238

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Abstract. We construct an invariant J M of integral homology spheres M with values in a completion Z[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU (2) Witten-ReshetikhinTuraev invariant τ ζ (M ) of M at ζ. Thus J M unifies all the SU (2) WittenReshetikhin-Turaev invariants of M . As a consequence, τ ζ (M ) is an algebraic integer. Moreover, it follows that τ ζ (M ) as a function on ζ behaves like an "analytic function" defined on the set of roots of unity. That is, the τ ζ (M ) for all roots of unity are determined by a "Taylor expansion" at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, τ ζ (M ) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q = 1.

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“…(This result generalizes a conjecture of Melvin and Morton [61], proved in [2].) More precisely, Rozansky proved that J MMR K has the following power series expansion…”

Section: Remarks On Knot Invariantssupporting

confidence: 80%

A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres

Habiro

2007

Invent. math.

238

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“…It was conjectured by Melvin and Morton [88] and later proved by Rozansky [29], and by Bar-Natan and Garoufalidis [89], that the coefficients D m,n (K) in the expansion (6.3) have the following properties 20 ,…”

Section: Jones Polynomial Asmentioning

confidence: 97%

Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial

Gukov

2005

Commun. Math. Phys.

272

521

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We study three-dimensional Chern-Simons theory with complex gauge group SL(2,C), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-MortonRozansky conjecture, and suggest an intriguing connection between the SL(2,C) partition function and the colored Jones polynomial.

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“…About 10 years ago, Melvin-Morton and Rozansky independently conjectured a relation among the limiting behavior of the colored Jones function of a knot and its Alexander polynomial (see Corollary 1.5), [MM,Ro1,Ro2]. D. Bar-Natan and the first author reduced the conjecture about knot invariants to a statement about their combinatorial weight systems, and then proved it for all weight systems that come from semisimple Lie algebras using combinatorial Lie algebraic methods, [BG].…”

mentioning

confidence: 99%

A non-commutative formula for the colored Jones function

Garoufalidis

Loebl

2006

Math. Ann.

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Abstract. The colored Jones function of a knot is a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. It has been understood in terms of representations of quantum groups and Witten gave an intrinsic quantum field theory interpretation of the colored Jones function as the expectation value of Wilson loops of a 3-dimensional gauge theory, the Chern-Simons theory. We present the colored Jones function as an evaluation of the inverse of a non-commutative fermionic partition function. This result is in the form familiar in quantum field theory, namely the inverse of a generalized determinant. Our formula also reveals a direct relation between the Alexander polynomial and the colored Jones function of a knot and immediately implies the extensively studied Melvin-Morton-Rozansky conjecture, first proved by Bar-Natan and the first author about ten years ago.

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The coloured Jones function

Cited by 91 publications

References 5 publications

A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres

A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres

Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial

A non-commutative formula for the colored Jones function

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