Contact Heegaard Splittings
It was asked how one might see the contact Heegaard splitting associated to the JVHM open book on T3. Two pages of an open book form a Heegaard surface that is convex with respect to the induced contact structure and the binding is the dividing set.
(Recall that the presentation being used for T3 is a hexagonal prism with opposite sides identified.)
We can squish it down to one side to make one of the handlebodies more apparent. The binding goes to the red dividing curves.
Squishing it down to the other side would’ve given the same picture as this last one, but with a half rotation around the horizontal hexagon.
So what makes this a contact Heegaard splitting rather than just a splitting with convex Heegaard surface?
A contact handlebody is a regular neighborhood of a Legendrian graph. A contact Heegaard splitting is a Heegaard splitting that divides the contact manifold into two contact handlebodies. (Definitions due to Giroux.) It turns out that given a contact manifold, a contact Heegaard splitting is essentially the same as a supporting open book.
The contact structure as viewed here has the contact planes go through one full rotation in the vertical direction while just translating in the horizontal directions. Each of the two Legendrian graphs is a vertical circle and three horizontal ones. In each graph of these pictures, the three vertical rods are all identified together, and so the three vertices of the graph are all valence 4.
Here’s a few more views. Or you can just check out the entire set of pictures.