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January 2019

Quantum Gravity Lesson 27

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:


Or looking at the 2-circulants:

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. 

Quantum Gravity Lesson 26

Pipe dreams are related to our favourite Hopf algebras in this recent paper ...

Pipe dreams and Hopf algebras

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We count the number of strings in a pipe dream by including the missing elbow pieces. For example, the second diagram for 12345 has two missing pieces: at the top and the bottom right. Now these seven strings are matched to seven sides on the polygon that will be chorded to give the four dimensional associahedron (that you now know and love).

Generalising the descent from the 3d permutohedron to the parity cube (particle charges) we have, for different k, the polytopes in the picture below. See how the parity cube morphs into the full permutohedron, with six signs.


Quantum Gravity Lesson 25


Meszaros likes to use sideways tree graphs, but I have drawn a few binary rooted trees to show that a tree is just a tree. As you can see, the numbers on the nodes count the number of right moving edges off the vertex. These are our usual integral Postnikov coordinates for the MacLane pentagon within the tetractys diagram. Yes, pipe dreams will relate braid type diagrams to trees! Pretty fundamental stuff.

Observe how the sideways trees look like they should order the nodes on a binary tree, as for the permutohedra, but they don’t, as the centre upper tree shows. It is pipe dreams that represent permutations as string diagrams. 


The dream is a ‘matrix picture’. Allowed tiles are called crosses or elbows, for obvious reasons, and there is more than one representation for a permutation. Forget the purple tree. A three letter permutation needs three nodes, like we had yesterday:

0DB27CD9-2026-423E-BE5D-1958F3277CE2These nodes come from a technical Young diagram which counts crossing tiles on all representations such that each flip map from S2 only gets one crossing tile. 

Karola Meszaros publications

A citation or not?

An email I sent today:


You should be ashamed of yourself, not properly acknowledging me in your new arxiv paper. I taught all these people you mention about polytopes, and I also discussed them with you, as you know. The number one reason I am continuously abused is because of sexist pigs like you.

Homeless in Christchurch

Truini’s arxiv paper