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Towards a symplectic Khovanov homology for links in fibered $3$-manifolds

Vincent Colin, Ko Honda, Yin Tian

Abstract

The goal of this paper is twofold: (i) define a symplectic Khovanov type homology for a transverse link in a fibered closed $3$-manifold $M$ (with an auxiliary choice of a homotopy class of loops that intersect each fiber once) and (ii) give conjectural combinatorial dga descriptions of surface categories that appear in (i). These dgas are higher-dimensional analogs of the strands algebras in bordered Heegaard Floer homology, due to Lipshitz-Ozsváth-Thurston \cite{LOT}.

Towards a symplectic Khovanov homology for links in fibered $3$-manifolds

Abstract

The goal of this paper is twofold: (i) define a symplectic Khovanov type homology for a transverse link in a fibered closed -manifold (with an auxiliary choice of a homotopy class of loops that intersect each fiber once) and (ii) give conjectural combinatorial dga descriptions of surface categories that appear in (i). These dgas are higher-dimensional analogs of the strands algebras in bordered Heegaard Floer homology, due to Lipshitz-Ozsváth-Thurston \cite{LOT}.

Paper Structure

This paper contains 48 sections, 40 theorems, 159 equations, 27 figures.

Table of Contents

  1. Introduction
  2. Symplectic Khovanov type homology of a transverse link in a fibered $3$-manifold
  3. Combinatorial description
  4. The local model
  5. $A_n$-singularity case
  6. The surface case
  7. Wrapped higher-dimensional Heegaard Floer homology
  8. Definition of the fully wrapped $A_{\infty}$-algebra and $A_{\infty}$-bimodules
  9. The $A_{\infty}$-operations
  10. Handleslides
  11. First handleslide model
  12. Second handleslide model
  13. The handleslide theorem
  14. Symplectic Khovanov homology of a transverse link in a fibered $3$-manifold
  15. The Lefschetz fibration $\Sigma =W_{S,n}$
  16. ...and 33 more sections

Key Result

Theorem 1.1.1

The triangulated envelope $\overline {\mathcal{R}}^\star (S,n,{\bf a})$ of $\mathcal{R}^\star (S,n,{\bf a})$ does not depend on the choice of ${\bf a}$, when $\star$ is a partial wrapping with respect to $\tau$ or a full wrapping.

Figures (27)

  • Figure 1: The red arc is $a$, the projection of the Lagrangian to the base $D^2$, and the two marks on the boundary form the stop.
  • Figure 2: The categorical action of $\check{\mathcal{E}}$ given by adding a Lagrangian near the stop. The red arc is the projection of the Lagrangian to the base $D^2$.
  • Figure 3: The $A_\infty$-base $D_m$ of the symplectic fibration $D_m\times \widehat{W}$.
  • Figure 4: Decomposing the Lagrangian resolution of $p_c$ into two cobordisms. The arrows indicate a surgery operation and $\simeq$ indicates the trace of a Legendrian isotopy.
  • Figure 5: The Legendrians $\partial a_1$, $\partial b_1$, and $\partial \widetilde{a}_1$ in $\partial W$ and the chords $c$, $d$, and $e$.
  • ...and 22 more figures

Theorems & Definitions (103)

  • Remark 1.0.1
  • Theorem 1.1.1
  • Theorem 1.1.2
  • Remark 1.1.3
  • Theorem 1.1.4
  • Theorem 1.1.5
  • Conjecture 1.1.6
  • Remark 1.1.7
  • Remark 1.1.8
  • Theorem 1.2.1
  • ...and 93 more