Evolution method and "differential hierarchy" of colored knot polynomials

TL;DR

This paper introduces an evolution method for analyzing colored knot polynomials, revealing a hierarchical structure in their behavior across various knot families and extending previous observations to superpolynomials.

Contribution

It develops a new evolution approach for colored knot polynomials and uncovers a hierarchical structure, including new examples like the double braid, enhancing understanding of knot invariants.

Findings

Reproduces known polynomial results for simple knots.

Introduces the double braid as a new example.

Shows the hierarchical structure extends to superpolynomials.

Abstract

We consider braids with repeating patterns inside arbitrary knots which provides a multi-parametric family of knots, depending on the "evolution" parameter, which controls the number of repetitions. The dependence of knot (super)polynomials on such evolution parameters is very easy to find. We apply this evolution method to study of the families of knots and links which include the cases with just two parallel and anti-parallel strands in the braid, like the ordinary twist and 2-strand torus knots/links and counter-oriented 2-strand links. When the answers were available before, they are immediately reproduced, and an essentially new example is added of the "double braid", which is a combination of parallel and anti-parallel 2-strand braids. This study helps us to reveal with the full clarity and partly investigate a mysterious hierarchical structure of the colored HOMFLY polynomials, at least, in (anti)symmetric representations, which extends the original observation for the figure-eight knot to many (presumably all) knots. We demonstrate that this structure is typically respected by the t-deformation to the superpolynomials.