Augmentations, annuli, and Alexander polynomials

TL;DR

This paper links the Alexander polynomial of a knot to the augmentation variety in knot contact homology, using counts of holomorphic annuli and Floer strips in symplectic geometry.

Contribution

It provides a novel expression of the Alexander polynomial in terms of the augmentation variety and holomorphic annuli, connecting knot invariants with symplectic geometry.

Findings

Expressed Alexander polynomial via augmentation variety and derivatives.

Connected knot contact homology with Floer theory and holomorphic annuli.

Derived a geometric interpretation of the Alexander polynomial.

Abstract

The augmentation variety of a knot is the locus, in the 3-dimensional coefficient space of the knot contact homology dg-algebra, where the algebra admits a unital chain map to the complex numbers. We explain how to express the Alexander polynomial of a knot in terms of the augmentation variety: it is the exponential of the integral of a ratio of two partial derivatives. The expression is derived from a description of the Alexander polynomial as a count of Floer strips and holomorphic annuli, in the cotangent bundle of Euclidean 3-space, stretching between a Lagrangian with the topology of the knot complement and the zero-section, and from a description of the boundary of the moduli space of such annuli with one positive puncture.