Upper and lower branched surface

Description

If we look at a Transverse veering triangulation and its Upper and lower train tracks, then they form a graph in a manifold. We imagine this graph being the skeleton of a pair of branched surfaces, the Upper and lower branched surface.

Definition

Let $M$ be an oriented $3$-manifold equipped with a Transverse veering triangulation $\mathcal{V}$. Let $t$ be a Veering tetrahedron in $\mathcal{V}$. Looking at a tetrahedron from above the upper and lower train tracks might look as follows Note that for the upper train track the lower side of a tetrahedron always has a Large branch whereas the upper side doesn’t. It is impossible for the upper side to have a large branch as the veering structure forbids two tetrahedron stacked on top of each other. It follows from the image (A) below, that the train tracks bound a quadrilinear and two triangles. The two surfaces obtained this way are called the upper and lower branched surfaces $B^t,B_t$ in $t$. The union over all $t$ gives us the upper and lower branched surfaces $B^{\mathcal{V}},B_{\mathcal{V}}$

Properties

"Valence" of the surface

We study $B^{\mathcal{V}}$. As we move from bottom two top, a sector splits into two sectors, every time we glue two sides. This means that every point in an edge is adjacent to $3$ sectors. Every vertex point is adjacent to $6$ sectors. The same applies for $B_{\mathcal{V}}$

Dual position

The upper branch can be moved into dual position by a small upward isotopy. If this is done¸ then every tetrahedron contains exactly one point and every face is pierced by exactly one branch locus. This reminds me of the proof with the charges.