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Augmentations, annuli, and Alexander polynomials

Luís Diogo, Tobias Ekholm

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.

Augmentations, annuli, and Alexander polynomials

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.

Paper Structure

This paper contains 39 sections, 45 theorems, 142 equations, 19 figures.

Table of Contents

  1. Introduction
  2. The Alexander polynomial and gradient flows
  3. Dynamical definition
  4. Morse theory calculation
  5. Flow loops, flow lines, and holomorphic curves
  6. Models of $L_{K}$ and $M_{K}$
  7. Holomorphic disks
  8. Holomorphic annuli and strips
  9. Orientations of moduli spaces
  10. Disks with corners
  11. Annuli
  12. Knot contact homology and augmentations
  13. Geometry of coefficients in the dg-algebra
  14. The augmentation variety and polynomial
  15. Analysis of boundaries of moduli spaces
  16. ...and 24 more sections

Key Result

Theorem 1.1

For any knot $K$, the full augmentation variety $\tilde{V}_{K}$ is smooth and 2-dimensional in a neighborhood $\tilde{V}^{0}_{K}$ (with respect to the standard metric topology on $(\mathbb{C}^{\ast})^{3}\times \mathbb{C}^{|\boldsymbol{a}|}$) of $\boldsymbol{\epsilon_0}$. Also, $d_{\boldsymbol{\epsil

Figures (19)

  • Figure 1: Schematic picture of the graph of $\widehat{\phi}$. The arrows indicate where the summands of $C_{\ast}(\widetilde{Y}_{K};\mu)$ are supported (when restricted to a fundamental domain for the action of $\mathbb{Z}$ on $\widetilde{Y}_K$ by deck transformations).
  • Figure 2: Completing $u$ into a closed surface to determine the $Q$-power
  • Figure 3: Elliptic boundary splitting
  • Figure 4: Horizontally, a family of disks crossing a rigid annulus. Diagonally, hyperbolic boundary splitting of a family of annuli with a boundary puncture
  • Figure 5: The boundary of $\mathcal{M}_{\mathrm{an}}(y)$
  • ...and 14 more figures

Theorems & Definitions (125)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 2.1: About the ambiguity of the Alexander polynomial
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 115 more