Invariants of Colored Links

Akutsu, Yasuhiro; Deguchi, Testuo; Ohtsuki, Tatsuyuki

doi:10.1142/s0218216592000094

Cited by 81 publications

(200 citation statements)

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“…Even though eventually this possibility may play an important róle in the connection with the SL(2,C) Chern-Simons theory, it seems less obvious at present. See however [83], which may be relevant here. I wish to thank D. Thurston for pointing out this reference and for helpful discussions on related topics.…”

Section: The A-polynomial and The Generalized Volume Conjecturementioning

confidence: 99%

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

Gukov

2005

Commun. Math. Phys.

265

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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Section: The A-polynomial and The Generalized Volume Conjecturementioning

confidence: 99%

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

Gukov

2005

Commun. Math. Phys.

265

521

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“…The condition (3) follows from Corollary 3.2. To prove (1), it suffices to prove J B ∈ (Ũ ev q )⊗ 3 . Using Figure 4.4, we obtain…”

Section: 2mentioning

confidence: 99%

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

Habiro

2007

Invent. math.

237

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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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“…More interestingly, this paper will contain two examples where the genuine quantum dimensions vanish but the "fake quantum dimensions" are non-zero and lead to non-trivial link invariants. The first of these examples recover the hierarchy of invariants defined by Akutsu, Deguchi and Ohtsuki [1], using a regularize of the Markov trace and nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of knot (see [16]).…”

Section: Introductionmentioning

confidence: 82%

“…These invariants associate a variable to each component of the link. There are only a handful of such invariants including the multivariable Alexander polynomial and the ones defined in [1]. All of these invariants are related to the invariants defined in this paper.…”

Section: Introductionmentioning

confidence: 99%