State models and the jones polynomial

Kauffman, Louis H.

doi:10.1016/0040-9383(87)90009-7

Cited by 1,151 publications

(735 citation statements)

References 5 publications

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“…what can be tested in the other two available examples in table (3.2) -7 4 and 9 5 . This conjecture provides a double-evolution matrix, which through (2.2) gives the entries of dX dȲSXȲ .…”

Section: Factorization Of Differential Expansion For Double-braidsmentioning

confidence: 99%

“…The coefficient c in (7.22) should be a quantization of 5, but it should contain odd powers of q, thus it can not be [5]. In (7.21) its analogue is a quantization of 4, which is made out of even powers -and it is [2] 2 .…”

Section: Other Floorsmentioning

confidence: 99%

“…Construction of knot polynomials [1][2][3][4][5][6][7][8] is currently at the front-line of modern theoretical physics, because this is an exactly solvable problem in quantum field theory, which remains unsolved for years, despite tremendous effort by many distinguished researchers. Far ago it was reformulated in terms of Chern-Simons theory [9]- [17], where (at least for the knots in S 3 ) it is basically reduced to a free-field calculation, and a formal answer is provided [18]- [32] in terms of representation theory of quantum groups.…”

Section: Introductionmentioning

confidence: 99%

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Factorization of differential expansion for antiparallel double-braid knots

Морозов

2016

J. High Energ. Phys.

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Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution -that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, generalizing the one-parametric family of twist knots. In the case of rectangular representations R = [r s ] we found an evidence that the corresponding differential expansion miraculously factorizes and can be obtained from that for the twist knots. This reduces the problem of rectangular exclusive Racah to constructing the answers for just a few twist knots. We develop a recent conjecture on the structure of differential expansion for the simplest members of this family (the trefoil and the figure-eight knot) and provide the exhaustive answer for the first unknown case of R = [33]. The answer includes HOMFLY of arbitrary twist and double-braid knots and Racah matricesS and S -what allows to calculate [33]-colored polynomials for arbitrary arborescent (double-fat) knots. For generic rectangular representations fully described are only the contributions of the single-floor pyramids. One step still remains to be done.

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Section: Factorization Of Differential Expansion For Double-braidsmentioning

confidence: 99%

Section: Other Floorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Factorization of differential expansion for antiparallel double-braid knots

Морозов

2016

J. High Energ. Phys.

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“…where U c is the trivial c component link diagram [18]. The Jones polynomial J 2 (K; t) of a knot K with a diagram D is defined as…”

Section: The Colored Jones Polynomial and Its Asymptotic Behaviormentioning

confidence: 99%

The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot

Murakami

2014

Acta Math Vietnam

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Abstract. A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern-Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers. If the knot is hyperbolic, the representation can be regarded as a deformation of the holonomy representation that determines the complete hyperbolic structure. In this article we study a similar phenomenon when the knot is a twice-iterated torus knot. In this case, the asymptotic expansion of the colored Jones polynomial splits into sums and each summand is related to the Chern-Simons invariant and the Reidemeister torsion associated with a representation.Let J N (K; q) ∈ Z[q, q −1 ] be the colored Jones polynomial of a knot K in the three-sphere S 3 associated with the irreducible N -dimensional representation of the Lie algebra sl 2 (C) [14,19]. We normalize it so that J N (unknot; q) = 1. Note that J 2 (K; q) is the original Jones polynomial. R. Kashaev conjectured [16] that his knot invariant K N ∈ C introduced in [15] would grow exponentially with growth rate the volume of the knot complement S 3 \ K when the integer parameter N goes to the infinity if the knot K is hyperbolic. J. Murakami and the author [34] proved that Kashaev's invariant coincides with J N K; exp(2π √ −1/N ) and generalized Kashaev's conjecture to general knots.

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“…This has led to rewrite suitable combinations of Chern-Simons knot polynomials [5][6][7][8][9][10][11][12][13][14] as reformulated invariants possessing integrality structures [15][16][17][18]69] famously known as LMOV condition (see also [19][20][21] for the latest development). These integer invariants count the BPS states (spectra of M2 branes ending on M5 branes in M-theory compactified on the conifold [2,3]).…”

Section: Introductionmentioning

confidence: 99%

Checks of integrality properties in topological strings

Mironov¹,

Морозов²,

Морозов³

et al. 2017

J. High Energ. Phys.

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Tests of the integrality properties of a scalar operator in topological strings on a resolved conifold background or orientifold of conifold backgrounds have been performed for arborescent knots and some non-arborescent knots. The recent results on polynomials for those knots colored by SU(N ) and SO(N ) adjoint representations [1] are useful to verify Marino's integrality conjecture up to two boxes in the Young diagram. In this paper, we review the salient aspects of the integrality properties and tabulate explicitly for an arborescent knot and a link. In our knotebook website, we have put these results for over 100 prime knots available in Rolfsen table and some links. The first application of the obtained results, an observation of the Gaussian distribution of the LMOV invariants is also reported.

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State models and the jones polynomial

Cited by 1,151 publications

References 5 publications

Factorization of differential expansion for antiparallel double-braid knots

Factorization of differential expansion for antiparallel double-braid knots

The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot

Checks of integrality properties in topological strings

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