Using branched covers might be a nice way to resolve those pentagon collisions in my pictures as well!

Over on Google+, Roice Nelson and I were discussing possibly related tactics for avoiding collisions in the hyperbolic plane, although I was suggesting using the hyperbolic tiling by decagons and pentagons obtained by truncating the {5,5} tiling, whereas your branched cover is related to the {5,10} tiling:

]]>http://blogs.ams.org/visualinsight/2015/02/01/pentagon-decagon-packing/

and even more this one:

http://blogs.ams.org/visualinsight/2015/02/15/pentagon-decagon-branched-covering/

which was based on our attempts to “solve the problem” of the nonexistent tiling of the plane by regular pentagons and decagons. The study of this problem goes back to Kepler, so it’s nice to see your new twist on it!

]]>As for “twist” vs. “loop”, I don’t really like “twist” in this context. As with Hopf algebras, it’s sometimes nice to draw the *antipode* as a twist, and you can see from the weak Hopf algebra axioms that this isn’t the same as what I called a “loop”.

The next bit of the story is now up, by the way.

]]>(But I’ve never quite decided which sort of wiggle is a “zig” and which is a “zag”. Once when I was falling asleep I thought I’d figured it out: a ziggurat should have lots of zigs. But when I woke up that no longer made sense.)

On another silly terminological note, I urge you to use “twist” instead of “loop” since “loop” means something else, while “twist” or “framing twist” is widely used to mean what you’ve drawn above.

More importantly, I’m eager to hear where this story goes next!

]]>Indeed, I should have said that the “twists” or “loops” are built, in order, from:

- the “cap” (comultiplication of the unit)
- the braiding
- the “cup” (the counit applied to a product)

where the braiding is symmetric here, since I’m working in the symmetric monoidal category of vector spaces. (Of course, the diagrams also tell you how to do Hopf algebras is a braided monoidal category, but that’s another story.)

I should also have pointed out that the “cap” and “cup” here are not the ones people might expect from string diagrams for compact closed categories. Indeed, these other kinds of caps and cups might not exist in general, since a weak Hopf algebra needn’t be finite-dimensional. In a situation where you need both kinds of caps and cups (for example, if the *dual* of a finite dimensional weak Hopf algebra is also heavily involved) my notation has to be modified. Adding arrows to all the strands suffices for that purpose, where all of the diagrams in my post have downward arrows.

The zigzag laws don’t hold for my caps and cups! I guess adding arrows to the diagrams would also make it less tempting to pull zigzags straight without looking at the rules.

On the other hand, the **zigzagzig** and **zagzigzag** laws both *do* hold. One of these says that a “zigzagzig” can be reduced to a “zig”, and the other says a “zagzigzag” can be reduced to a “zag”. I hope you know what I mean. These follow from sticking another unit or counit onto the middle strand in the equation relating the comultiplication to the unit and in the equation relating the multiplication to the counit.

As you mentioned, it’s also nice to use a framing in these diagrams. Unfortunately, this still doesn’t make all of the allowed moves topologically obvious.

]]>One thing that’s confusing me is those two “twist” diagrams you draw, which are “essentially graphical representations of the target and source maps”. Clearly you want these to be different, but I’m confused about why.

At first I forgot how you defined the “cap” and “cup” from which you build these twists (together with the braiding). Now I remember, but I still don’t know if they obey the zigzag identities. If they don’t, you should warn us that not all topologically plausible moves are allowed. If they do, and your braiding is symmetric, it seems those two twists should be inverses of each other.

I’m fairly confused, but I hope you see my point. In a naive topological interpretation of your diagrams, you should be able to “pull straight” either twist and see that it’s the identity. In a less naive interpretation, the strings in your diagrams have a framing that can keep track of a 360 twist. Then one of your twist diagrams corresponds to a clockwise twist and the other corresponds to a counterclockwise twist.

Maybe you know the “Whitney trick” shown on page 15 of Kauffman’s book:

http://homepages.math.uic.edu/~kauffman/KFI.pdf

That’s the sort of thing that’s on my mind.

]]>Somehow along this other “X” I guess you’ve still got to break the tangent space of SO(4,1) into some particular copy of so(3,1) and a complement, so that the Cartan connection consists of an SO(3,1) connection plus a coframe field. What’s the “symmetry breaking” field in this case?

Anyway, to really understand this, what I would still like is a *geometric* description of how, after choosing a point to serve as the identity, any point in de Sitter space gives me an isometry of de Sitter space, and a geometric understanding of why these particular isometries form a subgroup of all of the isometries. I suspect that a really good geometric description of this should not involve talking about the Lie algebras.

But anyway, if you are writing stuff up about this de Sitter subgroup, and if you don’t already have such a geometric description, maybe we can figure it out after I read what you’ve got.

]]>http://theoreticalatlas.wordpress.com/2012/10/08/observer-space-cartan-gr/

]]>If I were trying to guess how this “X” works, I might start by breaking the Lie algebra into representations of SO(3,1):

and then exponentiating in the directions. This will locally give a 4d submanifold of SO(4,1) that looks like de Sitter space, but it sure won’t give me a subgroup, since isn’t a subalgebra. And also, it’ll only work locally. So, I guess you must mean something different.

]]>