I have been attempting to draw the Cayley graph of the braid group $$ B_3 = \langle a, b \mid aba=bab \rangle$$ and I obtained something that almost seems too good to be true; here is a picture. This might require some explanation:
- The vertices of the graph in question are the little circles. Note that the intersections that do not involve circles are not vertices, they are non-planarity artifacts.
- Multiplying by the generators $a,b$ corresponds to moving to the top left, to the top right, respectively. Similarly, multiplying by their inverses $a^{-1}, b^{-1}$ corresponds to moving to the bottom right, bottom left, respectively.
- Of course, the picture does not contain the full graph, but is rather one step in an iteration that yields the full graph after countably many steps. Here is (a zoomed-in version of) one further step in the iteration.
Question. Is the graph explained above the Cayley graph of the braid group $B_3$ on three strands?
Ideas and strategies on how to proceed are also greatly appreciated.
