Colored HOMFLY polynomials for the pretzel knots and links

Abstract

With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g+1 two strand braids, parallel or antiparallel, and depend on g+1 integer numbers. We demonstrate that they possess a pronounced new structure: are decomposed into a sum of a product of g+1 elementary polynomials, which are obtained from the evolution eigenvalues by rotation with the help of rescaled SU_q(N) Racah matrix, for which we provide an explicit expression. The generalized pretzel family contains many mutants, undistinguishable by symmetric HOMFLY polynomials, hence, the extension of our results to non-symmetric representations R is a challenging open problem. To this end, a non-trivial generalization of the suggested formula can be conjectured for entire family with arbitrary g and R.

26 pages + tables of pretzel knots up to 10 crossings

References in corpus (57)

• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
• Matrix factorizations and link homology II
• Torus knots and mirror symmetry
• Complete Set of Cut-and-Join Operators in Hurwitz-Kontsevich Theory
• Soft matrix models and Chern-Simons partition functions
• On the characteristic and deformation varieties of a knot
• Knot Homology from Refined Chern-Simons Theory
• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• On Link Invariants and Topological String Amplitudes
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links III : Compact Semi-simple Group
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links
• Chern-Simons Theory in the Temporal Gauge and Knot Invariants through the Universal Quantum R-Matrix
• Algebra of differential operators associated with Young diagrams
• On the Hecke algebras and the colored HOMFLY polynomial
• Super-A-polynomial for knots and BPS states
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
• Evolution method and "differential hierarchy" of colored knot polynomials
• Character expansion for HOMFLY polynomials. I. Integrability and difference equations
• Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations
• Knot invariants from rational conformal field theories
• 3d analogs of Argyres-Douglas theories and knot homologies
• SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial
• Racah coefficients and extended HOMFLY polynomials for all 5-, 6- and 7-strand braids
• Colored knot polynomials for Pretzel knots and links of arbitrary genus
• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
• Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux
• Multiplicity-free quantum 6j-symbols for U_q(sl_N)
• Super-A-polynomials for Twist Knots
• Chirality of Knots $9_{42}$ and $10_{71}$ and Chern-Simons Theory
• Differential hierarchy and additional grading of knot polynomials
• On colored HOMFLY polynomials for twist knots
• Knot polynomials in the first non-symmetric representation
• Knot invariants from Virasoro related representation and pretzel knots
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial
• Towards matrix model representation of HOMFLY polynomials
• DAHA and iterated torus knots
• Equations on knot polynomials and 3d/5d duality
• Colored Kauffman Homology and Super-A-polynomials
• A note on colored HOMFLY polynomials for hyperbolic knots from WZW models
• The Superpolynomial for Knot Homologies
• Interplay between MacDonald and Hall-Littlewood expansions of extended torus superpolynomials
• Reformulated invariants for non-torus knots and links
• Colored HOMFLY polynomial via skein theory
• Link polynomial calculus and the AENV conjecture
• Wall Crossing Invariants: from quantum mechanics to knots
• Kontsevich integral for knots and Vassiliev invariants
• Vassiliev Invariants for Links from Chern-Simons Perturbation Theory
• Cabling procedure for the colored HOMFLY polynomials
• Special colored Superpolynomials and their representation-dependence
• The first-order deviation of superpolynomial in an arbitrary representation from the special polynomial
• Super-A-polynomial
• Invariants of links of Conway type
• Matrix integral expansion of colored Jones polynomials for figure-eight knot

Cited by in corpus (58)

• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Correlators in tensor models from character calculus
• Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots
• On universal knot polynomials
• Towards effective topological field theory for knots
• Tabulating knot polynomials for arborescent knots
• Knot invariants from Virasoro related representation and pretzel knots
• Colored knot polynomials. HOMFLY in representation [2,1]
• Towards topological quantum computer
• Racah matrices and hidden integrability in evolution of knots
• Differential expansion and rectangular HOMFLY for the figure eight knot
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Quantum Racah matrices and 3-strand braids in irreps R with |R|=4
• Factorization of differential expansion for antiparallel double-braid knots
• On rectangular HOMFLY for twist knots
• Tangle blocks in the theory of link invariants
• Nimble evolution for pretzel Khovanov polynomials
• Differential expansion for link polynomials
• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Multiplicity-free $U_q(s{l}_N)$ 6-j symbols: relations, asymptotics, symmetries
• Eigenvalue hypothesis for multi-strand braids
• Are there p-adic knot invariants?
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities
• On R-matrix approaches to knot invariants
• Rectangular superpolynomials for the figure-eight knot
• Eigenvalue conjecture and colored Alexander polynomials
• Perspectives of differential expansion
• Quantum $\mathcal{R}$-matrices as universal qubit gates
• Distinguishing Mutant Knots
• Checks of integrality properties in topological strings
• Extension of KNTZ trick to non-rectangular representations
• New symmetries for the $U_q(s{l}_N)$ 6-j symbols from the Eigenvalue conjecture
• Orthogonal Polynomials in Mathematical Physics
• Quantum Racah matrices up to level 3 and multicolored link invariants
• On exclusive Racah matrices $\bar{S}$ for rectangular representations
• Quantum Racah matrices and 3-strand braids in representation [3,3]
• Can tangle calculus be applicable to hyperpolynomials?
• Gaussian distribution of LMOV numbers
• On $6j$-symbols for symmetric representations of $U_q({\mathfrak{su}}_N)$
• HOMFLY for twist knots and exclusive Racah matrices in representation [333]
• On the block structure of the quantum R-matrix in the three-strand braids
• Towards tangle calculus for Khovanov polynomials
• KNTZ trick from arborescent calculus and the structure of differential expansion
• Planar decomposition of bipartite HOMFLY polynomials in symmetric representations
• Pentad and triangular structures behind the Racah matrices
• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• Wilson Loop Invariants from $W_N$ Conformal Blocks
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• SU(2)/SL(2) knot invariants and KS monodromies
• Evolution properties of the knot's defect
• Vassiliev invariants for pretzel knots
• Knots and signal transmission in topological quantum devices
• Entangled states from arborescent knots
• Matrix model and dimensions at hypercube vertices
• Entangling gates from cabling of knots
• HOMFLY Polynomials of Pretzel Knots
• On ambiguity in knot polynomials for virtual knots
• On an alternative stratification of knots