Revisiting the Melvin-Morton-Rozansky Expansion, or There and Back Again

TL;DR

This paper explores reconstructing advanced knot invariants from the Alexander polynomial through a process of rewriting, quantization, and deformation, confirming existing conjectures and deriving new formulas for colored superpolynomials.

Contribution

It introduces a novel method to reconstruct colored HOMFLY-PT polynomials, superpolynomials, and Z invariants from Alexander polynomial, confirming conjectures and producing new formulas.

Findings

Rederived conjectural invariants for various knots.

Proved consistency with Melvin-Morton-Rozansky theorem.

Derived new formulas for colored superpolynomials.

Abstract

Alexander polynomial arises in the leading term of a semi-classical Melvin-Morton-Rozansky expansion of colored knot polynomials. In this work, following the opposite direction, we propose how to reconstruct colored HOMFLY-PT polynomials, superpolynomials, and newly introduced $\widehat{Z}$ invariants for some knot complements, from an appropriate rewriting, quantization and deformation of Alexander polynomial. Along this route we rederive conjectural expressions for the above mentioned invariants for various knots obtained recently, thereby proving their consistency with the Melvin-Morton-Rozansky theorem, and derive new formulae for colored superpolynomials unknown before. For a given knot, depending on certain choices, our reconstruction leads to equivalent expressions, which are either cyclotomic, or encode certain features of HOMFLY-PT homology and the knots-quivers correspondence.