On the defect and stability of differential expansion

TL;DR

This paper investigates the stability of colored knot polynomials in Chern-Simons theory, revealing universal properties and introducing the concept of defect related to the Alexander polynomial.

Contribution

It uncovers the stability phenomenon of knot polynomial coefficients and introduces the defect as a new universal classifying parameter for knots.

Findings

Coefficients of knot polynomials stabilize for large symmetric representations.

Existence of universality classes of knots characterized by a single integer called defect.

Defect is related to the power of the Alexander polynomial.

Abstract

Empirical analysis of many colored knot polynomials, made possible by recent computational advances in Chern-Simons theory, reveals their stability: for any given negative N and any given knot the set of coefficients of the polynomial in r-th symmetric representation does not change with r, if it is large enough. This fact reflects the non-trivial and previously unknown properties of the differential expansion, and it turns out that from this point of view there are universality classes of knots, characterized by a single integer, which we call defect, and which is in fact related to the power of Alexander polynomial.