Lectures on Knot Homology and Quantum Curves

TL;DR

This paper explores the connections between knot homologies, quantum curves, and classical invariants like the A-polynomial, proposing a deformation that categorifies the Generalized Volume Conjecture and distinguishes mutant knots.

Contribution

It introduces a novel deformation of the A-polynomial's algebraic curve that links knot homologies with quantum curves and categorifies the Generalized Volume Conjecture.

Findings

Established a direct relation between Khovanov homology and the A-polynomial.

Proposed a deformation that categorifies the Generalized Volume Conjecture.

Showed the deformation distinguishes mutant knots.

Abstract

Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this interpretation allows one to pose questions that would not have been asked otherwise, such as, "Is there a direct relation between Khovanov homology and the A-polynomial of a knot?" We will explain that the answer to this question is "yes," and introduce a certain deformation of the planar algebraic curve defined by the zero locus of the A-polynomial. This novel deformation leads to a categorified version of the Generalized Volume Conjecture that completely describes the "color behavior" of the colored sl(2) knot homology, and eventually to a similar version for the colored HOMFLY homology. Furthermore, this deformation is strong enough to distinguish mutants, and its most interesting properties include relations to knot contact homology and knot Floer homology.