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Holomorphic polygons and the bordered Heegaard Floer homology of link complements

Thomas Hockenhull

TL;DR

This work constructs Poly$(L,Λ)$, an $ ext{A}_ fty$ multi-module over the torus algebra $\\mathcal{T}$, defined via counts of holomorphic polygons in Heegaard multi-diagrams for link complements. It proves Poly$(L,Λ)$ is quasi-isomorphic to the bordered-sutured invariant $BSA(S^3 - L, Λ)$, connecting bordered sutured Floer theory to link Floer homology in a way that will enable computations from CFL$^-(L)$ in future work. The methodology blends splayed/bordered diagrams, detailed moduli-space analysis, splicings of Reeb chords, and neck-stretching/anchoring techniques to control end behavior and gluing, culminating in an $ ext{A}_ fty$-equivalence result. The framework clarifies how to decompose and reassemble invariants under gluings, with implications for satellite constructions and mapping-cone formulas in knot and link Floer theory.

Abstract

We describe the construction of an $\mathcal{A}_\infty$ multi-module in terms of counts of holomorphic polygons in a series Heegaard multi-diagrams. We show that this is quasi-isomorphic to the type-A bordered-sutured invariant of a link complement with a view to calculating, in the sequel, these invariants in terms of the link Floer homology of the corresponding link.

Holomorphic polygons and the bordered Heegaard Floer homology of link complements

TL;DR

This work constructs Poly, an multi-module over the torus algebra , defined via counts of holomorphic polygons in Heegaard multi-diagrams for link complements. It proves Poly is quasi-isomorphic to the bordered-sutured invariant , connecting bordered sutured Floer theory to link Floer homology in a way that will enable computations from CFL in future work. The methodology blends splayed/bordered diagrams, detailed moduli-space analysis, splicings of Reeb chords, and neck-stretching/anchoring techniques to control end behavior and gluing, culminating in an -equivalence result. The framework clarifies how to decompose and reassemble invariants under gluings, with implications for satellite constructions and mapping-cone formulas in knot and link Floer theory.

Abstract

We describe the construction of an multi-module in terms of counts of holomorphic polygons in a series Heegaard multi-diagrams. We show that this is quasi-isomorphic to the type-A bordered-sutured invariant of a link complement with a view to calculating, in the sequel, these invariants in terms of the link Floer homology of the corresponding link.

Paper Structure

This paper contains 32 sections, 46 theorems, 152 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Organisation
  4. Acknowledgements
  5. Algebraic background
  6. The torus algebra
  7. $\mathcal{A}_\infty$ multi-modules
  8. Diagrams
  9. Heegaard multi-diagrams
  10. Bordered multi-diagrams
  11. Converting between diagrams
  12. Domains
  13. Splaying domains
  14. Moduli spaces of holomorphic curves and polygons
  15. Complex structures on polygons
  16. ...and 17 more sections

Key Result

Theorem 1

The invariant $\mathrm{Poly}(L, \Lambda)$ is quasi-isomorphic to the bordered-sutured multi-module ${{BSA}}(S^3 - L, \Lambda)$.

Figures (18)

  • Figure 1: The behaviour of the hamiltonian isotopy $H(x,t)$, where $t = 0$ is shown in red and $t = \varepsilon$ in orange.
  • Figure 2: Stabilisation data for a $4$-pointed basic Heegaard diagram. Here, the ${\pmb{\alpha}}$ curves are shown in red and the ${\pmb{\beta}}$ in blue.
  • Figure 3: The bordered stabilisation of a basic Heegaard diagram at one pair of basepoints
  • Figure 4: The closed stabilisation of a basic Heegaard diagram at one pair of basepoints $w_i, z_i$, where the idempotent $\iota = \iota_{\delta_1} \otimes \cdots \otimes \iota_{\delta_m}$ satisfies $\delta_i = m$
  • Figure 5: The closed stabilisation of a basic Heegaard diagram at one pair of basepoints $w_i, z_i$, where the idempotent $\iota = \iota_{\delta_1} \otimes \cdots \otimes \iota_{\delta_m}$ satisfies $\delta_i = l$.
  • ...and 13 more figures

Theorems & Definitions (134)

  • Theorem 1
  • Remark 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Proposition 9
  • proof
  • ...and 124 more