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Lattice homology, formality, and plumbed L-space links

Maciej Borodzik, Beibei Liu, Ian Zemke

Abstract

We define a link lattice complex for plumbed links, generalizing constructions of Ozsváth, Stipsicz and Szabó, and of Gorsky and Némethi. We prove that for all plumbed links in rational homology 3-spheres, the link lattice complex is homotopy equivalent to the link Floer complex as an $A_\infty$-module. Additionally, we prove that the link Floer complex of a plumbed L-space link is a free resolution of its homology. As a consequence, we give an algorithm to compute the link Floer complexes of plumbed L-space links, in particular of algebraic links, from their multivariable Alexander polynomial.

Lattice homology, formality, and plumbed L-space links

Abstract

We define a link lattice complex for plumbed links, generalizing constructions of Ozsváth, Stipsicz and Szabó, and of Gorsky and Némethi. We prove that for all plumbed links in rational homology 3-spheres, the link lattice complex is homotopy equivalent to the link Floer complex as an -module. Additionally, we prove that the link Floer complex of a plumbed L-space link is a free resolution of its homology. As a consequence, we give an algorithm to compute the link Floer complexes of plumbed L-space links, in particular of algebraic links, from their multivariable Alexander polynomial.
Paper Structure (44 sections, 44 theorems, 234 equations, 8 figures)

This paper contains 44 sections, 44 theorems, 234 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. The link lattice complex
  4. Algebraic and plumbed L-space links
  5. Gorsky and Némethi's link lattice homology
  6. Structure of the paper
  7. Algebraic background
  8. $A_\infty$-modules
  9. The homological perturbation lemma
  10. Free resolutions and $A_\infty$-actions
  11. Example of non-formal chain complexes
  12. Plumbed manifolds and plumbed links
  13. Review of plumbed manifolds
  14. Plumbed manifolds and resolutions of analytic singularities
  15. Embedded resolutions
  16. ...and 29 more sections

Key Result

Theorem 1.1

Suppose that $\Gamma$ is a plumbing link diagram which is a tree, and write $(Y,L)$ for the associated 3-manifold and link. If $Y$ is a rational homology sphere, then $\boldsymbol{\mathbf{\mathcal{C\!F\!L}}}(Y,L)$ is homotopy equivalent to $\mathbb{CFL}(\Gamma,V_\uparrow)$ as an absolutely graded $A

Figures (8)

  • Figure 1.1: The knot and link Floer complexes of $T(3,4)$, $T(2,2)$ and $T(2,4)$. Each dot denotes a generator in a free basis. The horizontal direction indicates the grading of the free resolution.
  • Figure 2.1: The maps appearing in the homological perturbation lemma for $A_\infty$-modules. The notation is introduced in LOTBordered*Section 2. Shortly, a single arrow represents an element of ${}_\mathcal{A} M$, while a double arrow represents an element of $\bigoplus_{i}\mathcal{A}^{\otimes i}$.
  • Figure 3.1: Resolution graph of the $E_8$ singularity. All vertices correspond to spheres with self-intersection $-2$. The meaning of components $A$ and $B$ is explained in the text.
  • Figure 3.2: Plumbing graph of the second knot in Proposition \ref{['prop:non_isotopic']}(b). All weights that are not explicitly marked have value $-2$.
  • Figure 3.3: The two links of CDG represented as graph links of EisenbudNeumannGraphLinks.
  • ...and 3 more figures

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.5
  • Remark 1.6
  • Corollary 1.7
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Remark 2.4
  • ...and 89 more