Mixed tori in contact surgery diagrams

TL;DR

The paper develops a diagrammatic approach to the symplectic JSJ decomposition for exact/weak fillings of contact 3-manifolds arising from Legendrian surgery, reducing filling classification to simpler split components via mixed tori and round 1-handle operations. It provides a general theorem tying fillings of a largely complicated manifold to fillings of a finite family of simpler pieces (M_k, ξ_k) through symplectic round 1-handles, and demonstrates the method by recovering known classifications for virtually overtwisted lens spaces and torus bundles. Extending the framework to plumbed 3-manifolds, the authors introduce wrapped-up graphs and Stein diagrams to translate plumbing data into Legendrian surgery diagrams, reducing the problem to maximal consistent subgraphs Γ_i. The work offers a practical algorithm for broad classes of plumbed manifolds and connects diagrammatic techniques with classical results, enabling systematic classification of exact/weak fillings in new settings.

Abstract

We develop a diagrammatic framework for applying the symplectic JSJ decomposition to exact/weak symplectic fillings of 3-dimensional contact manifolds. Namely, we apply the symplectic JSJ decomposition to a contact surgery diagram for some $(Y,ζ)$, producing a finite collection of contact manifolds, also described diagrammatically, whose exact/weak symplectic fillings determine those of $(Y,ζ)$. We apply this technique to recover known symplectic filling classifications for certain lens spaces and torus bundles, and also to provide an algorithm for classifying the exact/weak symplectic fillings of a large class of plumbed 3-manifolds.

Figures (9)

• Figure 1: The link $L=\Lambda_1\sqcup\cdots\sqcup\Lambda_n$ is an inconsistent chain which passes over the belt sphere $S$ of a symplectic 1-handle. The chain is inconsistent because, while both $\Lambda_1$ and $\Lambda_n$ are negatively-stabilized, the product of the linking numbers $\ell k(\Lambda_1,\Lambda_2),\ldots,\ell k(\Lambda_{n-1},\Lambda_n)$ is $-1$.
• Figure 2: In \ref{['subsec:background:spheres']} we describe a canonical splitting of the component $\Lambda_k$ of $L$ which intersects $S$ into a link $\Lambda_k^+\sqcup\Lambda_k^-$. We use this splitting to define a self-linking number in the case where $\Lambda_1$ and $\Lambda_2$ coincide.
• Figure 3: By writing $(M,\xi)$ as the connected sum $(M\# S^3,\xi\#\xi_{\mathrm{std}})$, any Legendrian knot $S_+(S_-(L))$ which has been stabilized both positively and negatively can be realized as an inconsistent chain passing over a surgery sphere.
• Figure 4: In the upper left is a fully-decorated good plumbing graph $\Gamma$, with undecorated edges assumed to carry a positive sign. Each vertex $v$ is labeled with $b(v)$ to its left (in black) and $r(v)$ to its right (in blue). Applying \ref{['thm:fillings-of-round-surgery']} to $\Gamma$ produces a subgraph, and iterating the process leaves us with two maximal consistent subgraphs of $\Gamma$.
• Figure 5: There are $\ell$ ways to break a subchain $C\subseteq C_{p,q}$ of length $\ell$, and each results in a link on which Legendrian surgery produces disjoint union of lens spaces. In each case, a two-component link $\Lambda_k^+\sqcup\Lambda_k^-$ survives to this disjoint union, depicted here in grey.

Theorems & Definitions (20)

• Theorem 1.1: christian2018jsj
• Remark 1.2
• Theorem 1.3
• Theorem 1.4: christian2018jsj
• Corollary 1.5: christian2018jsj
• proof : Proof of \ref{['thm:mixed-stabilizations']}
• Theorem 1.6: etnyre2021symplecticchristian2023some
• Theorem 1.7: christian2021symplectic
• Theorem 1.8
• Proposition 3.1