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

2012/03/31 by H. Itoyama, А. Миронов, A. Mironov +3 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebra over a field #Algebraic structures and combinatorial models #Antisymmetric relation #Combinatorics #Geometric and Algebraic Topology #HOMFLY polynomial #Knot (papermaking) #Mathematical analysis #Mathematical physics #Mathematics #Matrix polynomial #Polynomial #Pure mathematics #hep-th #math.GT #math.QA

paper · pdf · doi:10.1007/jhep07(2012)131

published as Journal of High Energy Physics Volume 2012, Number 7 (2012), 131 · 14 pages

arxiv created 2012/06/12 · openalex publication_date 2012/07/01 · arxiv updated 2012/08/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

Explicit 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 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.

Citations

• An enumeration of knots and links, and some of their algebraic properties
• Knot Homology from Refined Chern-Simons Theory
• Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots
• The volume conjecture, perturbative knot invariants, and recursion relations for topological strings
• Framed knots at large N
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• Quantum field theory and the Jones polynomial
• Three-dimensional Chern-Simons theory as a theory of knots and links
• Torus HOMFLYPT as the Hall–Littlewood polynomials
• Characteristic Forms and Geometric Invariants
• A-polynomial, B-model, and quantization
• Topological invariants of knots and links
• State models and the jones polynomial
• On the relation between the A-polynomial and the Jones polynomial

Cited by

• Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
• Nontorus link from topological vertex