On geometric bases for quantum A-polynomials of knots

TL;DR

This paper introduces a geometric approach to derive quantum A-polynomials of knots, linking classical and quantum invariants through contact geometry and quantization, with a focus on the trefoil knot and potential for broader generalizations.

Contribution

It presents a simplified geometric method to derive quantum A-polynomials, making these techniques more accessible and extending the framework to colored Jones polynomials for knots.

Findings

Geometric derivation of Ward identities in Chern-Simons theory.

Simplification of quantum A-polynomial calculations using Kauffman calculus.

Potential for extending methods beyond the trefoil knot and Jones polynomials.

Abstract

A simple geometric way is suggested to derive the Ward identities in the Chern-Simons theory, also known as quantum $A$- and $C$-polynomials for knots. In quasi-classical limit it is closely related to the well publicized augmentation theory and contact geometry. Quantization allows to present it in much simpler terms, what could make these techniques available to a broader audience. To avoid overloading of the presentation, only the case of the colored Jones polynomial for the trefoil knot is considered, though various generalizations are straightforward. Restriction to solely Jones polynomials (rather than full HOMFLY-PT) is related to a serious simplification, provided by the use of Kauffman calculus. Going beyond looks realistic, however it remains a problem, both challenging and promising.