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

Abstract

Explicit answer is given for the HOMFLY polynomial of the figure eight knot $4_1$ 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 σ_R = σ_{[1]}^{|R|} 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 A-polynomial") in the representation variable p. Simple symmetry transformation provides also a formula for arbitrary antisymmetric (fundamental) representation R=[1^p], 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.

14 pages

References in corpus (40)

• Knot Invariants and Topological Strings
• All genus correlation functions for the hermitian 1-matrix model
• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
• Integrability and Matrix Models
• Khovanov-Rozansky Homology and Topological Strings
• Hermitean matrix model free energy: Feynman graph technique for all genera
• Knots, links and branes at large N
• Torus knots and mirror symmetry
• Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model
• Knot Homology from Refined Chern-Simons Theory
• Polynomial invariants for torus knots and topological strings
• The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link
• Framed knots at large N
• The colored Jones function is q-holonomic
• On Link Invariants and Topological String Amplitudes
• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• $2d$ gravity and matrix models. I. $2d$ gravity
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links III : Compact Semi-simple Group
• Instantons and Merons in Matrix Models
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links
• On the Hecke algebras and the colored HOMFLY polynomial
• Matrix eigenvalue model: Feynman graph technique for all genera
• BGWM as Second Constituent of Complex Matrix Model
• Super-A-polynomial for knots and BPS states
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• A-polynomial, B-model, and Quantization
• Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
• Quantum Dilogarithm as a 6j-Symbol
• Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots
• M-Theory of Matrix Models
• Character expansion for HOMFLY polynomials. I. Integrability and difference equations
• Knot invariants from rational conformal field theories
• Symplectic invariants, Virasoro constraints and Givental decomposition
• Partition Functions of Matrix Models as the First Special Functions of String Theory. II. Kontsevich Model
• Chern-Simons Invariants of Torus Links
• The Superpolynomial for Knot Homologies
• Interplay between MacDonald and Hall-Littlewood expansions of extended torus superpolynomials
• SU(N) quantum Racah coefficients & non-torus links
• Torus HOMFLY as the Hall-Littlewood Polynomials
• On the relation between the A-polynomial and the Jones polynomial

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• 3d analogs of Argyres-Douglas theories and knot homologies
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• Colored knot polynomials for Pretzel knots and links of arbitrary genus
• Refined Chern-Simons Theory and Topological String
• On universal knot polynomials
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• Differential hierarchy and additional grading of knot polynomials
• On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions
• On colored HOMFLY polynomials for twist knots
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• Tabulating knot polynomials for arborescent knots
• Colored knot polynomials. HOMFLY in representation [2,1]
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
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• Towards matrix model representation of HOMFLY polynomials
• Equations on knot polynomials and 3d/5d duality
• A note on colored HOMFLY polynomials for hyperbolic knots from WZW models
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