Relative Open Books in T^2 x I
Here are some relative open books in . They’re depicted in rectangular solids where you need to identify the left with the right and the top with the bottom. The pages of the open books meet the two torus boundary components (the front and the back) in curves. I’ve drawn “meridians” on them to suggest how to regard the monodromies as being the identity on the boundary of the page. Set in a bit from the boundary are marks that help to indicate the dynamics of the page. (Again, these are animated gifs.)
This first one is a relative open book for a “basic slice”. Refer to Section 4.2 and thereabouts of VanHorn-Morris’s dissertation (pdf). (This and the previous open book post take much inspiration from there.)
View the page as a once-punctured annulus with one positive Dehn twist around the puncture. The puncture becomes the binding.
Geometrically, notice that the pages here are all one single form that gets translated to the right. Not doing any Dehn twists along the annular boundary components means the marks on the pages follow the meridians.
It may seem odd that one of the annular boundary components is “vertical” while the other is “slanted”. But they both intersect the meridians once, the meridians are parallel, and they intersect each other once in the projection to the factor. Below it’s shown with another more symmetric presentation. The page is there too with the blue curve indicating a positive Dehn twist. (Identify the left and right to get the punctured annulus.)
Choosing different meridians on the boundaries is tantamount to doing Dehn twists along the boundaries. Below we have a once-punctured annulus page with a positive Dehn twist around the puncture and a negative Dehn twist around each annular boundary component. Notice how the marks move relative to the new meridians.
Next we’ll stack two together.
Take two copies of the first open book presented above. With only doing rigid movements (rotations, translations), we can stack them together in two ways.
In one way, their meridians match up nicely and we obtain a relative open book for that is a twice punctured annulus with a positive Dehn twist around each puncture.
In the other way, the meridians don’t match up. We can actually see their adjacent marks twisting relative to each other, a negative Dehn twist between them. The page here is a twice punctured annulus with a positive Dehn twist around each puncture and a negative Dehn twists along a curve that separates them.
This last open book is actually a positive Hopf stabilization of the very first open book. The monodromy takes an arc connecting the two punctures to a disjoint arc that veers to the right, handing you a positive Hopf band. (Perhaps I could make this more explicit.) Alternatively, one could use a clever application of the Lantern Relation as Goodman does in his Lemma 6.1, Figure 10 (arXiv).
Here are the two open books side by side.