### M Theory Lesson 144

The 9 faced associahedron in $\mathbb{R}^{3}$ keeps popping up in M Theory. Today we'll turn it into a pair of pants, with the three discs at the boundary corresponding to the three squares of the polytope, which were occasionally marked with crossings of a trefoil knot. The numbers labelling the real axis (which are a bit hard to read) are $-1$, $0$, $\frac{1}{2}$, $1$, $2$ and $\infty$. The first image is an extension of the Riemann sphere of lesson 62 to a hexagon on the real axis. The second image is a Grothendieck ribbon graph associated to the j invariant. Note that the ribbons pass through $-1$, $\frac{1}{2}$ and $2$ on the real axis.

By splitting the ribbon into six pieces on the pair of pants, marked with a trivalent vertex on the back and front, and attaching vertices to the nodes of the projection onto the plane, we find exactly 14 vertices, six pentagons and three squares, describing the associahedron. This might just be a bit of fun, until we look at what happens when we glue four of these pants together to form a genus 3 surface. By adding vertices on the squares from each side of the gluing, the pentagons are turned into heptagons, and we get a 24 heptagon tiling of the Klein quartic. Who said operads weren't useful?

By splitting the ribbon into six pieces on the pair of pants, marked with a trivalent vertex on the back and front, and attaching vertices to the nodes of the projection onto the plane, we find exactly 14 vertices, six pentagons and three squares, describing the associahedron. This might just be a bit of fun, until we look at what happens when we glue four of these pants together to form a genus 3 surface. By adding vertices on the squares from each side of the gluing, the pentagons are turned into heptagons, and we get a 24 heptagon tiling of the Klein quartic. Who said operads weren't useful?

## 5 Comments:

01 12 08

Thanks Kea. Glad to see you back. How interesting that this associahedron can be decomposed into my favorite pair of pants and secondly looking at the knot crossings. Thanks for this perspective;)

Thanks for mentioning the Klein Quartic.

I have a web page at

http://tony5m17h.net/KQphys.html

that talks about a Klein Quartic physics model.

If you want a pdf file, it is at

http://tony5m17h.net/KleinQP.pdf

but the first image will not move around as an animated image.

Tony Smith

Hi Tony, Could your dynamic and static representations of the Klein Quartic be classified as both:

a - a genus 3 torus and

b - some type of MÃ¶bius-like strip?

doug, as I said on my web page,

the animated image of the Klein Quartic was done by Greg Egan (and I found out about it from John Baez's web site). Greg Egan's web page at

http://www.gregegan.net/SCIENCE/KleinQuartic/KleinQuartic.html

has a lot of information and a link to a John Baez web page at

http://math.ucr.edu/home/baez/klein.html

which in turn has a link to my web site for one of its images.

To answer your questions in your comment:

Greg Egan (on the above-cited page) says:

"...Klein's quartic curve is a surface of genus 3, which is to say that it is like a 3-holed torus. ... it has the local geometry of the hyperbolic plane ...".

As to orientability, I think that (unlike the Klein bottle and the Moebius strip) the Klein Quartic is orientable.

Tony Smith

PS - A very nice book is "The Eightfold Way - The Beauty of Klein's Quartic Curve", ed. by Silvio Levy, MSRI Publication 35 (Cambridge 1999).

Thanks for the nice link, Tony. I've come across this page of yours before. I see that you mention Segal and the twistor group. Part of what I'm looking for here is a (less group theoretic) geometric alternative to the 4:1 cover by SU(2,2), such as the 4:1 gluing for the Klein surface. After all, category theory doesn't much mind what 'models' one works with, since it is the meta-algebra level that is physical.

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