Whenever you see permutation diagrams, you should be thinking of the permutations underlying the braid diagrams for Standard Model states. The neutrino only uses a B3 diagram, because it has no charge. If we relabel the boxes in a pipe dream, we see how 312 (a left or right handed particle) only has one possible tree diagram. Turning a 1-circulant into a 1-circulant:
On three nodes (n = 3) for S3 we naturally consider the associator map between words of length 3. Thus pipe dreams justify the idea that the 27 word basis is doubled to left and right bracket sets, corresponding directly to chirality for neutrinos.
Allowing for over and under braid crossings, we would have three tile types, including the pipe dream elbows.