Defect and degree of the Alexander polynomial

TL;DR

This paper proves that the defect of knot polynomials correlates with the degree of the Alexander polynomial in q, providing new insights into their structure and a complete set of C-polynomials for defect zero knots.

Contribution

It establishes the conjecture linking defect to the Alexander polynomial degree and analyzes antiparallel descendants of 2-strand torus knots for all defect values.

Findings

Defect equals the degree of the Alexander polynomial in q^{ extpm 2}.

Confirmed the conjecture for defect zero knots, including twist knots.

Provided a complete set of C-polynomials for defect zero symmetric Alexander polynomials.

Abstract

Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be alternatively described as the degree in $q^{\pm 2}$ of the fundamental Alexander polynomial, which formally corresponds to the case of no colors. We also pose a question if these Alexander polynomials can be arbitrary integer polynomials of a given degree. A first attempt to answer the latter question is a preliminary analysis of antiparallel descendants of the 2-strand torus knots, which provide a nice set of examples for all values of the defect. The answer turns out to be positive in the case of defect zero knots, what can be observed already in the case of twist knots. This proved conjecture also allows us to provide a complete set of $C$-polynomials for the symmetrically colored Alexander polynomials for defect zero. In this case, we achieve a complete separation of representation and knot variables.