## Lifting an open book to a cover

In his dissertation Jeremy Van Horn-Morris describes a certain open book for the Stein fillable contact structure on $T^3$. Its page is a thrice punctured torus. It’s conjectured that this open book realizes the binding number of this contact structure, i.e. there’s no open book with twice punctured torus that supports the contact structure. (See Etnyre & Ozbagci – Invariants of contact structures from open books.) Anyway, I wanted to take a look at it. (Update: But this isn’t it. The top and bottom need to glue together with a half rotation. To get the JVHM open book for $T^3$ there needs to be some Dehn twists between the levels.)

This uses the standard presentation of $T^3$ as a cube with opposite faces identified. This only shows one page of the open book.

To get a better picture of all the connectivity, we can take covers of $T^3$ and lift the open book.

More pictures after the break.

Yet more are on the Flickr page. Just hit any of these pictures to jump there.

So I suppose you could fill up $\mathbb{R}^3$ with these cubes to make an infinite periodic open book. It should support the standard contact structure there too.

I’ll have to revisit this: make a more symmetric one using a hexagonal prism as the fundamental domain for $T^3$, draw a few more pages, color the two sides different colors. (Update: Coloring the two sides shows something is amiss.)