Differential expansion for link polynomials

Abstract

The differential expansion is one of the key structures reflecting group theory properties of colored knot polynomials, which also becomes an important tool for evaluation of non-trivial Racah matrices. This makes highly desirable its extension from knots to links, which, however, requires knowledge of the $6j$-symbols, at least, for the simplest triples of non-coincident representations. Based on the recent achievements in this direction, we conjecture a shape of the differential expansion for symmetrically-colored links and provide a set of examples. Within this study, we use a special framing that is an unusual extension of the topological framing from knots to links. In the particular cases of Whitehead and Borromean rings links, the differential expansions are different from the previously discovered.

11 pages

References in corpus (42)

• Torus knots and mirror symmetry
• Soft matrix models and Chern-Simons partition functions
• 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
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links
• On the Hecke algebras and the colored HOMFLY polynomial
• 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
• Colored HOMFLY polynomials for the pretzel knots and links
• Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots
• Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations
• Character expansion for HOMFLY polynomials. I. Integrability and difference equations
• Knot invariants from rational conformal field theories
• Knots-quivers correspondence
• Colored knot polynomials for Pretzel knots and links of arbitrary genus
• Racah coefficients and extended HOMFLY polynomials for all 5-, 6- and 7-strand braids
• BPS states, knots and quivers
• On universal knot polynomials
• Colored HOMFLY polynomials as multiple sums over paths or standard Young tableaux
• Towards effective topological field theory for knots
• Chirality of Knots $9_{42}$ and $10_{71}$ and Chern-Simons Theory
• Topological Strings, D-Model, and Knot Contact Homology
• On colored HOMFLY polynomials for twist knots
• Knot polynomials in the first non-symmetric representation
• Tabulating knot polynomials for arborescent knots
• Colored knot polynomials. HOMFLY in representation [2,1]
• Colored HOMFLY polynomials that distinguish mutant knots
• Reformulated invariants for non-torus knots and links
• On the defect and stability of differential expansion
• Racah matrices and hidden integrability in evolution of knots
• Differential expansion and rectangular HOMFLY for the figure eight knot
• Homological algebra of knots and BPS states
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Quantum Racah matrices and 3-strand braids in irreps R with |R|=4
• On rectangular HOMFLY for twist knots
• Cabling procedure for the colored HOMFLY polynomials
• 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
• Quantum Racah matrices and 3-strand braids in representation [3,3]
• Colored HOMFLY polynomials from Chern-Simons theory

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• Hopf link invariants and integrable hierarchies
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• Distinguishing Mutant Knots
• Knot polynomials for twist satellites
• On Hopf-induced deformation of topological locus
• Quantum Racah matrices up to level 3 and multicolored link invariants
• A non-torus link from topological vertex
• Planar decomposition of the HOMFLY polynomial for bipartite knots and links
• Can tangle calculus be applicable to hyperpolynomials?
• HOMFLY for twist knots and exclusive Racah matrices in representation [333]
• Khovanov polynomials for satellites and asymptotic adjoint polynomials
• Difference of mutant knot invariants and their differential expansion
• Pentad and triangular structures behind the Racah matrices
• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• Evolution properties of the knot's defect
• Invariants of knots and links at roots of unity
• Hopf superpolynomial from topological vertices