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 (10)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
• Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations
• Colored knot polynomials for Pretzel knots and links of arbitrary genus
• BPS states, knots and quivers
• On colored HOMFLY polynomials for twist knots
• Reformulated invariants for non-torus knots and links
• On rectangular HOMFLY for twist knots
• On R-matrix approaches to knot invariants
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities