Differential expansion for link polynomials

TL;DR

This paper extends the differential expansion framework from knots to links, providing conjectures and examples for symmetrically-colored links, and explores new framing methods with specific applications to Whitehead and Borromean rings.

Contribution

It proposes a conjecture for the differential expansion of symmetrically-colored links and introduces a novel framing approach extending from knots to links.

Findings

Differential expansion for links differs from that for knots in specific cases.

New framing method extends topological framing to links.

Explicit examples include Whitehead and Borromean rings.

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.