A note on colored HOMFLY polynomials for hyperbolic knots from WZW models

Abstract

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 HOMFLY 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.

72 pages, 15 figures; v2: explicit HOMFLY polynomials for more knots included, refs. added and typos corrected; v3: Appendix with sample calculation added

References in corpus (19)

• Knot Invariants and Topological Strings
• Boundary Conditions in Rational Conformal Field Theories
• Chern-Simons Theory and Topological Strings
• Torus knots and mirror symmetry
• Polynomial invariants for torus knots and topological strings
• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• On Link Invariants and Topological String Amplitudes
• The geometry of WZW branes
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links III : Compact Semi-simple Group
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links
• Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations
• Multiplicity-free quantum 6j-symbols for U_q(sl_N)
• Super-A-polynomials for Twist Knots
• Knot polynomials in the first non-symmetric representation
• Large N duality, lagrangian cycles, and algebraic knots
• Knot Invariants from Topological Recursion on Augmentation Varieties
• U(N) Framed Links, Three-Manifold Invariants, and Topological Strings
• Lectures on Knot Homology and Quantum Curves
• Chern-Simons Gauge Theory: Ten Years After

Cited by in corpus (26)

• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Colored HOMFLY polynomials for the pretzel knots and links
• Tabulating knot polynomials for arborescent knots
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Factorization of differential expansion for antiparallel double-braid knots
• Tangle blocks in the theory of link invariants
• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Eigenvalue hypothesis for multi-strand braids
• Multiplicity-free $U_q(s{l}_N)$ 6-j symbols: relations, asymptotics, symmetries
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities
• Distinguishing Mutant Knots
• Perspectives of differential expansion
• New symmetries for the $U_q(s{l}_N)$ 6-j symbols from the Eigenvalue conjecture
• Checks of integrality properties in topological strings
• Planar decomposition of the HOMFLY polynomial for bipartite knots and links
• On the block structure of the quantum R-matrix in the three-strand braids
• Entanglement on multiple $S^2$ boundaries in Chern-Simons theory
• KNTZ trick from arborescent calculus and the structure of differential expansion
• Planar decomposition of bipartite HOMFLY polynomials in symmetric representations
• Wilson Loop Invariants from $W_N$ Conformal Blocks
• Interplay between symmetries of quantum 6-j symbols and the eigenvalue hypothesis
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• On knots, complements, and 6j-symbols
• Tug-the-hook symmetry for quantum 6j-symbols
• Multistrand Eigenvalue conjecture and Racah symmetries
• On an alternative stratification of knots