Based on the feedback I got from the Buttohole Your Braid tutorial, I realized that I needed to address the braid that makes the buttonhole a little more. It's a four element braid which is unusual in the kumihimo world. I learned this braid from Jacqui Carey's Creative Kumihimo, where it is classified as braid 4A.
Lay out your disk like the photo below. Note that the elements that are on opposite sides of the disk are offset.
Lift up the bottom element and bring it to the top.
Bring the other element down, and put the elements in the starting positions.
Repeat with the other set of elements.
Sometimes, the elements will twist apart instead of together. If they do this, pass the element that is moving down underneath the element moving up when you next move that pair.
And this is what the braid will look like when you are done.