A. Morozov

HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations

A bstractExplicit answer is given for the HOMFLY polynomial of the figure eight knot 41 in arbitrary symmetric representation R = [p]. It generalizes the old answers for p = 1 and 2 and the recently derived results for p = 3, 4, which are fully consistent with the Ooguri-Vafa conjecture. The answer can be considered as a quantization of the identity for the “special” polynomials (they define the leading asymptotics of HOMFLY at q = 1), and arises in a form, convenient for comparison with the representation of the Jones polynomials as sums of dilogarithm ratios. In particular, we construct a difference equation (“non-commutative

Character expansion for HOMFLY polynomials. I. Integrability and difference equations

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Cabling procedure for the colored HOMFLY polynomials

-polynomial”) in the representation variable p. Simple symmetry transformation provides also a formula for arbitrary antisymmetric (fundamental) representation R = [1p], which also passes some obvious checks. Also straightforward is a deformation from HOMFLY to superpolynomials. Further generalizations seem possible to arbitrary Young diagrams R, but these expressions are harder to test because of the lack of alternative results, even partial.

Colored HOMFLY Polynomials as Multiple Sums over Paths or Standard Young Tableaux

We suggest to associate with each knot the set of coefficients of its HOMFLY polynomial expansion into the Schur functions. For each braid representation of the knot these coefficients are defined unambiguously as certain combinations of the Racah symbols for the algebra SU_q. Then, the HOMFLY polynomials can be extended to the entire space of time-variables. The so extended HOMFLY polynomials are no longer knot invariants, they depend on the choice of the braid representation, but instead one can naturally discuss their explicit integrable properties. The generating functions of torus knot/link coefficients are turned to satisfy the Plucker relations and can be associated with tau-function of the KP hierarchy, while generic knots correspond to more involved systems. On the other hand, using the expansion into the Schur functions, one can immediately derive difference equations (A-polynomials) for knot polynomials which play a role of the string equation. This adds to the previously demonstrated use of these character decompositions for the study of beta-deformations from HOMFLY to superpolynomials.

Evolution method and “differential hierarchy” of colored knot polynomials

We discuss using the cabling procedure to calculate colored HOMFLY polynomials. We describe how it can be used and how the projectors and

EIGENVALUE HYPOTHESIS FOR RACAH MATRICES AND HOMFLY POLYNOMIALS FOR 3-STRAND KNOTS IN ANY SYMMETRIC AND ANTISYMMETRIC REPRESENTATIONS

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Special colored Superpolynomials and their representation-dependence

-matrices needed for this procedure can be found. The constructed matrix expressions for the projectors and

Colored knot polynomials: HOMFLY in representation [2, 1]

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First-order deviation of superpolynomial in an arbitrary representation from the special polynomial

-matrices in the fundamental representation allow calculating the HOMFLY polynomial in an arbitrary representation for an arbitrary knot. The computational algorithm can be used for the knots and links with ¦Q¦m ≤ 12, where m is the number of strands in a braid representation of the knot and ¦Q¦ is the number of boxes in the Young diagram of the representation. We also discuss the justification of the cabling procedure from the group theory standpoint, deriving expressions for the fundamental

Towards matrix model representation of HOMFLY polynomials

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