Arced Diagram

3D Cobordism

Definition.
Given pointed matched circles ${\mathfrak{Z}}_L$ and ${\mathfrak{Z}}_R$, an arced cobordism is a compact 3-manifold $Y$ with:

• Injection $\phi :(-F^o({\mathfrak{Z}}_L))\bigsqcup F^o({\mathfrak{Z}}_R)\to \partial Y$.
• A path $\gamma \subset \partial Y-\mathrm{Im}(\phi )$ so that $Y-(\mathrm{Im}(\phi )\cup nbhd(\gamma ))$ is a disk.

Example.
Mapping cylinders of Dehn twists.

Definition (Arced Heegaard diagram).
A tuple

$$\mathcal{H}=({\Sigma }_g,{\alpha }^{a,L},{\alpha }^{a,R},{\alpha }^c,\beta ,\underline{z})$$

with disjoint $\alpha$ sets, connected complements, and transverse intersections.

From $\mathcal{H}$ we can build an arced cobordism by thickening, attaching handles, and tracking the arc $\gamma$.

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Examples

• Mapping cylinders.
• Dehn twists.
• Drilling: going from arced diagrams to bordered diagrams.

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Elementary Moves

Theorem.
Every bordered 3-manifold is represented by some bordered Heegaard diagram. Every arced cobordism is represented by some arced Heegaard diagram.

Theorem (Uniqueness up to moves).
If $\mathcal{H}$ and ${\mathcal{H}}^{\prime }$ represent equivalent bordered 3-manifolds, then they are related by:

• Isotopies of $\alpha ,\beta$ curves,
• Handleslides (circles and arcs),
• Stabilizations/destabilizations.