On the formulae for the colored HOMFLY polynomials

Kawagoe, Kenichi

doi:10.48550/arxiv.1210.7574

Cited by 9 publications

(13 citation statements)

References 5 publications

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“…However, this property cannot be checked unless expressions beyond the multiplicity-free ones are obtained. With our conjectured quantum 6j-symbols (10), we can verify that they reproduce the HOMFLY polynomials colored by the symmetric representation of many non-torus knots [8,9], the Whitehead link and the Borromean rings [12,7] up to 4 boxes. Moreover, we compute colored HOMFLY polynomials of many knots and links which we tabulate in the companion paper [16].…”

Section: The Fundamental Weights the Relation To The Young Tableau Ca...mentioning

confidence: 62%

Multiplicity-free Quantum 6j-Symbols for $${U_q(\mathfrak{sl}_N)}$$ U q ( sl N )

Nawata¹,

Ramadevi

Zodinmawia

2013

Lett Math Phys

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We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6j-symbols for U q (sl N ). The expression is a natural generalization of the quantum 6j-symbols for U q (sl 2 ) obtained by Kirillov and Reshetikhin. Our conjectured form enables computation of colored HOMFLY polynomials for various knots and links carrying arbitrary symmetric representations.Mathematics Subject Classification (2010): 17B37, 17B81, 20G42, 81R50

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Section: The Fundamental Weights the Relation To The Young Tableau Ca...mentioning

confidence: 62%

Multiplicity-free Quantum 6j-Symbols for $${U_q(\mathfrak{sl}_N)}$$ U q ( sl N )

Nawata¹,

Ramadevi

Zodinmawia

2013

Lett Math Phys

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“…So the complex volume for hyperbolic knot 5 2 is 2.828122 + 3.02413 √ −1. By using the formula for SU(n) invariant of 5 2 in [11]. We have the following 2) invariants, i.e., the colored Jones polynomials, converge to the complex volume faster than the general SU(n) invariants.…”

Section: Examples Of Conjecture 13 Case Of Knot 5mentioning

confidence: 99%

“…As for part (ii) of Conjecture 1.4, we believe it also holds for hyperbolic links although we have only checked the case of knots. In the paper [11], K. Kawagoe first proposed a special case of the part (ii) of Conjecture 1.4 for a = n − 1 and s = 1.…”

Section: Introductionmentioning

confidence: 99%

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Volume conjecture for $SU(n)$-invariants

Chen,

Liu,

Zhu

2015

Preprint

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This paper discuss an intrinsic relation among congruent relations [1], cyclotomic expansion and Volume Conjecture for SU (n) invariants. Motivated by the congruent relations for SU (n) invariants obtained in our previous work [1], we study certain limits of the SU (n) invariants at various roots of unit. First, we prove a new symmetry property for the SU (n) invariants by using a symmetry of colored HOMFLYPT invariants. Then we propose some conjectural formulas including the cyclotomic expansion conjecture and volume conjecture for SU (n) invariants (specialization of colored HOM-FLYPT invariants). We also give the proofs of these conjectural formulas for the case of figure-eight knot.

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“…For example, to test the assertions of our recent work [19], it would be interesting to have HOMFLY polynomials for hyperbolic knots colored with Young diagrams with up to two rows at our disposal. There are some results on HOMFLY polynomials for certain hyperbolic knots colored with totally symmetric and/or anti-symmetric representations [20][21][22][23][24][25][26]. More recently, for certain classes of knots HOMFLY invariants for colorings with more general representations have explicitly been obtained in refs.…”

Section: Introductionmentioning

confidence: 99%

A Note on Colored HOMFLY Polynomials for Hyperbolic Knots from WZW Models

Gu¹,

Jockers²

2015

Commun. Math. Phys.

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Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY invariants in terms of crossing matrices of the underlying Wess-Zumino-Witten model. Our analysis extends previous works by incorporating non-trivial multiplicities for the primaries appearing in the crossing matrices, so as to describe colorings of HOMFLY invariants beyond the totally symmetric or anti-symmetric representations of SU(N). The crossing matrices directly relate to 6j-symbols of the quantum group U q su(N). We present powerful methods to calculate such quantum 6j-symbols for general N. This allows us to determine previously unknown colored HOMFLY polynomials for two-bridge hyperbolic knots. We give explicitly the HOM-FLY polynomials colored by the representation {2, 1} for two-bridge hyperbolic knots with up to eight crossings. Yet, the scope of application of our techniques goes beyond knot theory; e.g., our findings can be used to study correlators in Wess-Zumino-Witten conformal field theories or -in the limit to classical groups -to determine color factors for Yang Mills amplitudes. July, 20142 The choice of k 1 and n is slightly different from ref.[21] to ensure that the charge n is integral. The regularization of the unnormalized colored HOMFLY invariants is nonetheless the same.

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On the formulae for the colored HOMFLY polynomials

Cited by 9 publications

References 5 publications

Multiplicity-free Quantum 6j-Symbols for $${U_q(\mathfrak{sl}_N)}$$ U q ( sl N )

Multiplicity-free Quantum 6j-Symbols for $${U_q(\mathfrak{sl}_N)}$$ U q ( sl N )

Volume conjecture for $SU(n)$-invariants

A Note on Colored HOMFLY Polynomials for Hyperbolic Knots from WZW Models

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