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Quantum Heegaard diagrams and knot Floer Homology

Cristina Ana-Maria Anghel, András Juhász

TL;DR

The paper introduces quantum Heegaard diagrams and a quantum Alexander grading to provide a unified geometric framework for the Alexander and Jones polynomials. By constructing a two-variable graded Lagrangian intersection on a quantum Heegaard surface, the authors recover knot Floer homology as a categorification of the Alexander polynomial and derive an explicit intersection model for the Jones polynomial. The approach hinges on braid actions on the punctured disc, configuration-space Lagrangians, and two interwoven gradings that align with knot Floer theory and a q-decorated Heegaard structure. This yields a concrete, geometric pathway to categorify both invariants and to realize a common setting from which both polynomials and their categorifications can be studied. The trefoil example demonstrates the method by recovering Δ_K and J_K, along with the corresponding knot Floer homology, from explicit graded intersections on the quantum Heegaard surface.

Abstract

We construct a unification of the Alexander and Jones polynomials via a new geometric perspective, involving "quantum Heegaard surfaces". Then, we prove that it leads to knot Floer homology and opens avenues for explicit geometric categorifications for the Jones polynomial. This model codifies the difference between the geometry of the Alexander and Jones polynomials via a quantum deformation of the Alexander grading, constructed on the surface. We do so by introducing a new type of diagram, called "quantum Heegaard diagram" (encoding the knot complement) together with a "quantum Alexander grading". This grading is a refinement of the Alexander grading used in knot Floer homology. Then we define a graded Lagrangian intersection, in two variables, between concrete Lagrangian submanifolds arising from the curves of the diagram in the symmetric power of the quantum Heegaard surface. The two-variable intersection unifies and recovers the Jones and Alexander polynomials as two specialisations of coefficients. Moreover, this leads to knot Floer homology on one side and provides an intersection model for the Jones polynomial on the other. The surface and Lagrangian submanifolds are concretely constructed via braid actions on the punctured disc.

Quantum Heegaard diagrams and knot Floer Homology

TL;DR

The paper introduces quantum Heegaard diagrams and a quantum Alexander grading to provide a unified geometric framework for the Alexander and Jones polynomials. By constructing a two-variable graded Lagrangian intersection on a quantum Heegaard surface, the authors recover knot Floer homology as a categorification of the Alexander polynomial and derive an explicit intersection model for the Jones polynomial. The approach hinges on braid actions on the punctured disc, configuration-space Lagrangians, and two interwoven gradings that align with knot Floer theory and a q-decorated Heegaard structure. This yields a concrete, geometric pathway to categorify both invariants and to realize a common setting from which both polynomials and their categorifications can be studied. The trefoil example demonstrates the method by recovering Δ_K and J_K, along with the corresponding knot Floer homology, from explicit graded intersections on the quantum Heegaard surface.

Abstract

We construct a unification of the Alexander and Jones polynomials via a new geometric perspective, involving "quantum Heegaard surfaces". Then, we prove that it leads to knot Floer homology and opens avenues for explicit geometric categorifications for the Jones polynomial. This model codifies the difference between the geometry of the Alexander and Jones polynomials via a quantum deformation of the Alexander grading, constructed on the surface. We do so by introducing a new type of diagram, called "quantum Heegaard diagram" (encoding the knot complement) together with a "quantum Alexander grading". This grading is a refinement of the Alexander grading used in knot Floer homology. Then we define a graded Lagrangian intersection, in two variables, between concrete Lagrangian submanifolds arising from the curves of the diagram in the symmetric power of the quantum Heegaard surface. The two-variable intersection unifies and recovers the Jones and Alexander polynomials as two specialisations of coefficients. Moreover, this leads to knot Floer homology on one side and provides an intersection model for the Jones polynomial on the other. The surface and Lagrangian submanifolds are concretely constructed via braid actions on the punctured disc.
Paper Structure (26 sections, 17 theorems, 94 equations, 16 figures)

This paper contains 26 sections, 17 theorems, 94 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Main result.
  3. Topological models in the disc
  4. Graded intersection in the configuration space of the punctured disc
  5. Heegaard diagram from braid actions on the punctured disc
  6. Categorification corresponding to knot Floer homology
  7. Unifying the Jones and the Alexander polynomials via quantum Heegaard surfaces
  8. Jones polynomial via quantum Heegaard diagrams
  9. Unified topological model for the Jones and Alexander polynomials
  10. Parameters for the graded intersection $\Omega$
  11. Grading of the intersection points
  12. Topological unification of the Jones and Alexander polynomials
  13. Intersection model for the Alexander polynomial
  14. Heegaard diagram from the topological model in the disc
  15. Heegaard diagrams for knots
  16. ...and 11 more sections

Key Result

Theorem 1.2

Let $K$ be an oriented knot and $\beta_n \in B_n$ a braid such that $K=\hat{\beta}_n$. Then the Alexander polynomial Here, $\varepsilon_{\bar{x}}$ is the sign of the geometric intersection between $(\beta_n\cup \mathbb I)\mathscr S$ and $\mathscr T$ in the configuration space and $w(\beta_n)$ is the writhe of the braid $\beta_n$.

Figures (16)

  • Figure 1.1: Local system and submanifolds for the Alexander polynomial
  • Figure 1.2: Disc model and the associated Heegaard diagram. On the right, the boundaries of the discs with the same colours are identified.
  • Figure 1.3: Quantum Heegaard diagram and q-Alexander gradings
  • Figure 2.1: The loops $\sigma_i$ (left) and $\delta$ (right).
  • Figure 2.2: The loops $\tilde{\sigma}_1, \dots, \tilde{\sigma}_{n-l}$, $\tilde{\gamma}_1, \dots, \tilde{\gamma}_l$, and $\tilde{\delta}$ (left) and the local system $\phi$ (right).
  • ...and 11 more figures

Theorems & Definitions (60)

  • Definition 1.1: Alexander grading in the punctured disc model
  • Theorem 1.2: Intersection formula for the Alexander polynomial Anghel2024AIF
  • Definition 1.3: Heegaard surface from the punctured disc
  • Definition 1.4: Sets of curves on $\Sigma$
  • Theorem 1.5: Heegaard diagram from the disc model
  • Definition 1.6: The knot Floer complex from the closure of the braid model in the punctured disc
  • Theorem 1.7: Local system grading is the Alexander grading
  • Theorem 1.8: The categorification is knot Floer homology
  • Definition 1.9: Quantum Heegaard diagrams and q-Alexander gradings
  • Definition 1.10: Quantum Lagrangian intersection
  • ...and 50 more