Optimistic Limits of the Colored Jones Polynomials

Cho, Jinseok; Murakami, Jun

doi:10.4134/jkms.2013.50.3.641

Cited by 19 publications

(45 citation statements)

References 12 publications

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“…We remark that the four term triangulation is related to the Kashaev invariant and the five term triangulation to the colored Jones polyonomial as mentioned in the introduction. Each tetrahedron corresponds to a q-series or a quantum factorial term of the R-matrices ([Yy1], [CM1]).…”

Section: The Octahedral Decomposition Of a Knot Complementmentioning

confidence: 99%

“…The more general non-collapsed version is done in [CKK]. The 5-term complex volume formula using the colored Jones polynomial was obtained by Cho and Murakami [CM1]. Both of these are elegant explicit formulas for complex volume which actually are anticipated from the volume conjecture as a limit, and hence this pseudo-hyperbolic structure is naturally involved in the limit algebro-analytic geometry of quantum invariants of a knot.…”

Section: Introductionmentioning

confidence: 99%

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Octahedral developing of knot complement I: Pseudo-hyperbolic structure

Kim

Yoon

2018

Geom Dedicata

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It is known that a knot complement can be decomposed into ideal octahedra along a knot diagram. A solution to the gluing equations applied to this decomposition gives a pseudo-developing map of the knot complement, which will be called a pseudo-hyperbolic structure. In this paper, we study these in terms of segment and region variables which are motivated by the volume conjecture so that we can compute complex volumes of all the boundary parabolic representations explicitly. We investigate the octahedral developing and holonomy representation carefully, and obtain a concrete formula of Wirtinger generators for the representation and also of cusp shape. We demonstrate explicit solutions for T (2, N ) torus knots, J(N, M ) knots and also for other interesting knots as examples. Using these solutions we can observe the asymptotic behavior of complex volumes and cusp shapes of these knots. We note that this construction works for any knot or link, and reflects systematically both geometric properties of the knot complement and combinatorial aspects of the knot diagram.

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Section: The Octahedral Decomposition Of a Knot Complementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Octahedral developing of knot complement I: Pseudo-hyperbolic structure

Kim

Yoon

2018

Geom Dedicata

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“…On the other hand, the colored Jones polynomial was shown to be a generalization of the Kashaev invariant in [12], and the optimistic limit of the colored Jones polynomial was also developed in [13], [6], [7] and [1]. Especially, following the idea of [4], another potential function W (w 1 , .…”

Section: Introductionmentioning

confidence: 99%

Optimistic limit of the colored Jones polynomial and the existence of a solution

Cho¹

2015

Proc. Amer. Math. Soc.

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For the potential function of a link diagram induced by the optimistic limit of the colored Jones polynomial, we show the existence of a solution of the hyperbolicity equations by directly constructing it. This construction is based on the shadow-coloring of the conjugation quandle induced by a boundary-parabolic representation ρ : π 1 (L) → PSL(2, C). This gives us a very simple and combinatorial method to calculate the complex volume of ρ.

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“…The proof of this proposition is quite complicate and technical, so we refer [3] and [5]. The following lemma was stated and used in [3] without proof because it is almost trivial.…”

Section: Figure 4: Three Gluing Patternsmentioning

confidence: 99%

“…The Reidemeister triansformations of the potential function still depend on the orientation. As a matter of fact, it is possible to define the potential function of the un-oriented diagram using Section 3.2 of[5]. However, the formula will be redundantly complicate than the one defined in this article, so we do not introduce it.…”

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confidence: 99%

Reidemeister transformations of the potential function and the solution

Cho

Murakami

2017

J. Knot Theory Ramifications

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The potential function of the optimistic limit of the colored Jones polynomial and the construction of the solution of the hyperbolicity equations were defined in the authors' previous articles. In this article, we define the Reidemeister transformations of the potential function and the solution by the changes of them under the Reidemeister moves of the link diagram and show the explicit formulas. These two formulas enable us to see the changes of the complex volume formula under the Reidemeister moves. As an application, we can simply specify the discrete faithful representation of the link group by showing a link diagram and one geometric solution.

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Optimistic Limits of the Colored Jones Polynomials

Abstract: Abstract. We show that the optimistic limits of the colored Jones polynomials of the hyperbolic knots coincide with the optimistic limits of the Kashaev invariants modulo 4π 2 .

Cited by 19 publications

References 12 publications

Octahedral developing of knot complement I: Pseudo-hyperbolic structure

Octahedral developing of knot complement I: Pseudo-hyperbolic structure

Optimistic limit of the colored Jones polynomial and the existence of a solution

Reidemeister transformations of the potential function and the solution

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