
On Tuesday I'm supposed to talk with Lee Smolin about an idea he's been working on with Fotini Markopoulou and Sundance BilsonThompson. This idea relates the elementary particles in one generation of the Standard Model to certain 3strand framed braids:
1) Sundance O. BilsonThompson, A topological model of composite preons, available as hepph/0503213.
2) Sundance O. BilsonThompson, Fotini Markopoulou, and Lee Smolin, Quantum gravity and the Standard Model, hepth/0603022.
It's a very speculative idea: they've found some interesting relations, but nobody knows if these are coincidental or not.
Luckily, one of my hobbies is collecting mysterious relationships between basic mathematical objects and trying to figure out what's going on. So, I already happen to know a bunch of weird facts about 3strand braids. I figure I'll tell Smolin about this stuff. But if you don't mind, I'll practice on you!
So, today I'll try to tell a story connecting the 3strand braid group, the trefoil knot, rational tangles, the groups SL(2,Z) and PSL(2,Z), conformal field theory, and Monstrous Moonshine.
I've talked about some of these things before, but now I'll introduce some new puzzle pieces, which come from two places:
3) Imre Tuba and Hans Wenzl, Representations of the braid group B_{3} and of SL(2,Z), available as math.RT/9912013.
4) Terry Gannon, The algebraic meaning of genuszero, available as math.NT/0512248.
You could call it "a tale of two groups".
On the one hand, the 3strand braid group has generators
   \ /  A = /  / \    and
    \ / B =  /  / \   and the only relation is
ABA = BAB
otherwise known as the "third Reidemeister move":
      \ /   \ / /   / / \   / \  \ / \ /   / = /   / \ / \  \ /   \ / /   / / \   / \On other hand, the group SL(2,Z) consists of 2×2 integer matrices with determinant 1. It's important in number theory, complex analysis, string theory and other branches of pure mathematics. I've described some of its charms in "week125", "week229" and elsewhere.
These groups look pretty different at first. But, there's a homomorphism from B_{3} onto SL(2,Z)! It goes like this:
1 1 A > 0 1 1 0 B > 1 1Both these matrices describe "shears" in the plane. You may enjoy drawing these shears and visualizing the equation ABA = BAB in these terms. I did.
I would like to understand this better... and here are some clues.
The center of B_{3} is generated by the element (AB)^{3}. This element corresponds to a "full twist". In other words, it's the braid you get by hanging 3 strings from the ceiling, grabbing them all with one hand at the bottom, and giving them a full 360degree twist:
   \ /  /  A / \   \ /  / B  / \ \ /  /  A / \   \ /  / B  / \ \ /  /  A / \   \ /  / B  / \   This full twist gets sent to 1 in SL(2,Z):
1 0 (AB)^{3} > 0 1So, the double twist gets sent to the identity:
1 0 (AB)^{6} > 0 1In fact, Tuba and Wenzl say the double twist generates the kernel of our homomorphism from B_{3} to SL(2,Z). So, SL(2,Z) is isomorphic to the group of 3strand braids modulo double twists!
This reminds me of spinors... since you have to twist an electron around twice to get its wavefunction back to where it started. And indeed, SL(2,Z) is a subgroup of SL(2,C), which is the double cover of the Lorentz group. So, 3strand braids indeed act on the state space of a spin1/2 particle, with double twists acting trivially!
(For more on this, check out Trautman's work on "Pythagorean spinors" in "week196". There's also a version where we use integers mod 7, described in "week219".)
If instead we take 3strand braids modulo full twists, we get the socalled "modular group":
PSL(2,Z) = SL(2,Z)/{+1}
Now, SL(2,Z) is famous for being the "mapping class group" of the torus  that is, the group of orientationpreserving diffeomorphisms, modulo diffeomorphisms connected to the identity. Similary, PSL(2,Z) is famous for acting on the rational numbers together with a point at infinity by means of fractional linear transformations:
az + b z >  cz + dwhere a,b,c,d are integers and adbc = 1. The group PSL(2,Z) also acts on certain 2strand tangles called "rational tangles". In "week229", I told a nice story I heard from Michael Hutchings, explaining how these three facts fit together in a neat package.
But now let's combine those facts with the stuff I just said! Since PSL(2,Z) acts on rational tangles, and there's a homomorphism from B_{3} to PSL(2,Z), 3strand braids must act on rational tangles. How does that go?
There's an obvious guess, or two, or three, or four, but let's just work it out.
I just said that the 3strand braid A gets mapped to this shear:
1 1 A > 0 1In "week229" I said what this shear does to a rational tangle. It gives it a 180 degree twist at the bottom, like this:
               T  >  T      \ /   /   / \Next, Tuba and Wenzl point out that
0 1 ABA = BAB > 1 0which is a quarter turn. From "week229" you can see how this quarter turn acts on a rational tangle:
      ____     / \        T  >   T           \____/        It gives it a quarter turn!
From these facts, we can figure out what the braid B does to a rational tangle. So, let me do the calculation.
Scribble, scribble, curse and scribble.... Eureka!
Since we know what A does, and what ABA does, we can figure out what B must do. But, to make the answer look cute, I needed a sneaky fact about rational tangles, which is that A also acts like this:
  \ /   /   / \    T  >  T               This is proved in Goldman and Kauffman's paper cited in "week228". With the help of this, I can show B acts like this:
       ___     / \    /   T  > \  T   / \     \___/         And this is great! It means our action of 3strand braids on rational tangles is really easy to describe. Just take your tangle and let the upper left strand dangle down:
 ____  / \      T            To let a 3strand braid act on this, just attach it to the bottom of the picture!
(That's why there were four obvious guesses about this would work: one can easily imagine four variations on this trick, depending on which strand is the "odd man out"  here it's the upper right. It's just a matter of convention which we use, but my conventions give this.)
In fact, even the group of 4strand braids acts on rational tangles in an obvious way, but the 3strand braid group is enough for now.
Let me summarize. The 3strand braid group B_{3} acts on rational tangles in an obvious way. The subgroup that acts trivially is precisely the center of B_{3}, generated by the full twist. Using stuff from "week229", it follows that the quotient of B_{3} by its center acts on the projectivized rational homology of the torus. We thus get a topological explanation of why B_{3} mod its center is PSL(2,Z).
But there's more.
For starters, the 3strand braid group is also the fundamental group of S^{3} minus the trefoil knot!
And, S^{3} minus the trefoil knot is secretly the same as SL(2,R)/SL(2,Z)!
In fact, Terry Gannon writes that the 3strand braid group can be regarded as "the universal central extension of the modular group, and the universal symmetry of its modular forms". I'm not completely sure what that means, but here's part of what it means.
Just as PSL(2,C) is the Lorentz group in 4d spacetime, PSL(2,R) is the Lorentz group in 3d spacetime. This group has a double cover SL(2,R), which shows up when you study spinors. But, it also has a universal cover, which shows up when you study anyons. The universal cover has infinitely many sheets. And up in this universal cover, sitting over the subgroup SL(2,Z), we get... the 3strand braid group!
In math jargon, we have this commutative diagram where the rows are short exact sequences:
1 > Z > B_{3} > SL(2,Z) > 1       v v v 1 > Z > SL(2,R)^{~} > SL(2,R) > 1Here SL(2,R)^{~} is the universal cover of SL(2,R). Since π_{1}(SL(2,R)) = Z, this is a Zfold cover. You can describe this cover explicitly using the Maslov index, which is a formula that actually computes an integer for any loop in SL(2,R), or indeed any symplectic group.
But anyway, fiddling around with this diagram and the long exact sequence of homotopy groups for a fibration, you can show that indeed:
π_{1}(SL(2,R)/SL(2,Z)) = B_{3}.
This also follows from the fact that SL(2,R)/SL(2,Z) looks like S^{3} minus a trefoil.
Gannon believes that number theorists should think about all this stuff, since he thinks it's lurking behind that weird network of ideas called Monstrous Moonshine (see "week66").
And here's the basic reason why. I'll try to get this right....
Any rational conformal field theory has a "chiral algebra" A which acts on the leftmoving states. Mathematicians call this sort of thing a "vertex operator algebra". A representation of this on some vector space V is a space of states for the circle in some "sector" of our theory. Let's pick some state v in V. Then we can define a "onepoint function" where we take a Riemann surface with little disk cut out and insert this state on the boundary. This is a number, essentially the amplitude for a string in the give state to evolve like this Riemann surface says.
In fact, instead of chopping out a little disk it's nice to just remove a point  a "puncture", they call it. But, we get an ambiguous answer unless we pick coordinates at this point, or in the lingo of complex analysis, a choice of "uniforming parameter". Then our onepoint function becomes a function on the moduli space of Riemann surfaces equipped with a puncture and a choice of uniformizing parameter.
If we didn't have this uniformizing parameter to worry about, we'd just have the moduli space of tori equipped with a marked point, which is nothing but the usual moduli space of elliptic curves,
H/PSL(2,Z)
where H is the complex upper halfplane. Then our onepoint function would have nice transformation properties under PSL(2,Z).
But, with this uniformizing parameter to worry about, our onepoint function only has nice transformation properties under B_{3}. This is somehow supposed to be related to how B_{3} is the "universal central extension" of PSL(2,Z): in conformal field theory, all sorts of naive symmetries hold only up to a phase, so you have to replace various groups by central extensions thereof... and here that's what's happening to PSL(2,Z)!
That last paragraph was pretty vague. If I'm going to understand this better, either someone has to help me or I've got to read something like this:
5) Yongchang Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc 9 (1996), 237302. Also available at http://www.ams.org/jams/1996901/S0894034796001828/home.html
But I shouldn't need any conformal field theory to see how the moduli space of punctured tori with uniformizing parameter is related to the 3strand braid group! I bet this moduli space is X/B_{3} for some space X, or something like that. There's something simple at the bottom of all this, I'm sure.
Addenda: Another relation between the trefoil and the punctured torus: the trefoil has genus 1, meaning that it bounds a torus minus a disc embedded in R^{3}. You can see this in the lecture "Genus and knot sum" in this course on knot theory:
6) Brian Sanderson, The knot theory MA3F2 page, http://www.maths.warwick.ac.uk/~bjs/MA3F2page.html
This course also has material on rational tangles.
The fact that B_{3} is a central extension of PSL(2,Z) by Z, and the quantummechanical interpretation of a central extension in terms of phases, plays an important role here:
7) R. Voituriez, Random walks on the braid group B_{3} and magnetic translations in hyperbolic geometry, Nucl. Phys. B621 (2002), 675688. Also available as http://arxiv.org/abs/mathph/0103008.
Among other things, he points out that the homomorphism B_{3} → SL(2,Z) described above is the "Burau representation" of B_{3} evaluated at t = 1. In general, the Burau representation of B_{3} is given by:
t 1 A > 0 1 1 0 B > t t(Conventions differ, and this may not be the best, but it's the one he uses.) The Burau representation can also be used to define a knot invariant called the Alexander polynomial. I believe that with some work, one can use this to explain why Conway could calculate the rational number associated to a rational tangle in terms of the ratio of Alexander polynomials of two links associated to it, called its "numerator" and "denominator". In fact he computed this ratio of polynomials and then evaluate it at a special value of t  presumably the same special value we're seeing here (modulo differences in convention).
Another issue: I wrote
For starters, the 3strand braid group is also the fundamental group of S^{3} minus the trefoil knot!The first one is pretty easy to see; you start with the "Wirtinger presentation" of the fundamental group of S^{3} minus a trefoil, and show by a fun little calculation that this isomorphic to the braid group on 3 strands. A more conceptual proof would be very nice, though. (See "week261" for such a proof  and much more on all this stuff.)And, S^{3} minus the trefoil knot is secretly the same as SL(2,R)/SL(2,Z)!
What about the second one? Why is S^{3} minus the trefoil knot diffeomorphic to SL(2,R)/SL(2,Z)? Terry Gannon says so in his paper above, but doesn't say why. Some people asked about this, and eventually some people found some explanations. First of all, there's a proof on page 84 of this book:
8) John Milnor, Introduction to Algebraic Ktheory, Annals of Math. Studies 72, Princeton U. Press, Princeton, New Jersey, 1971.
Milnor credits it to Quillen. Joe Christy summarizes it below. I can't tell if this proof is essentially the same as another sketched below by Swiatoslaw Gal, which exhibits a diffeomorphism using functions called the Eisenstein series g_{2} and g_{3}. They are probably quite similar arguments.
Joe Christy writes:
I wouldn't be surprised if this was known to Seifert in the 30's, though I can't lay my hands on Seifert & Threfall at the moment to check. Likewise for Hirzebruch, Brieskorn, Pham & Milnor in the 60's in relation to singularities of complex hypersurfaces and exotic spheres. When I was learning topology in the 80's it was considered a warm up case of Thurston's Geometrization Program  the trefoil knot complement has PSL(2,R) geometric structure.Swiatoslaw Gal writes:In any case, peruse Milnor's Annals of Math Studies for concrete references. There is a (typically) elegant proof on p.84 of "Introduction to Algebraic Ktheory" [study 72], which Milnor credits to Quillen. It contains the missing piece of John's argument: introducing the Weierstrass Pfunction and remarking that the differential equation that it satisfies gives the diffeomorphism to S^{3}trefoil as the boundary of the pair (discriminant of diffeq, C^{2} = (P,P')space).
This point of view grows out of some observations of Zariski, fleshed out in "Singular Points of Complex Hypersurfaces" [study 61]. The geometric viewpoint is made explicit in the paper "On the Brieskorn Manifolds M(p,q,r)" in "Knots, Groups, and 3manifolds" [study 84].
It is also related to the intermediate case between the classical Platonic solids and John's favorite Platonic surface  the Klein quartic. By way of a hint, look to relate the trefoil, qua torus knot, the sevenvertex triangulation of the torus, and the dual hexagonal tiling of a (flat) Clifford torus in S^{3}.
Joe
In fact the isomorphism is a part of the modular theory:The quantity g_{2}^{3}  27 g_{3}^{2} is called the "discriminant" of the lattice L, and vanishes as the lattice squashes down to being degenerate, i.e. a discrete subgroup of C with one rather than two generators.Looking for
f: GL(2,R)/SL(2,Z) → C^{2}  {x^{2}=y^{3}}
(there is an obvious action of R+ on both sides:
M → tM for M in GL(2,R),
x → t^{6} x,
y → t^{4} y,and the quotient is what we want).
GL(2,R)/SL(2,Z) is a space of lattices in C. Such a lattice L has classical invariants
g_{2}(L) = 60 sum_{z in L'} z^{4},
and
g_{3}(L) = 140 sum_{z in L'} z^{6},
where L'=L{0}
The modular theory asserts that:
1. For every pain (g_{2},g_{3}) there exist a lattice L such that g_{2}(L) = g_{2} and g_{3}(L) = g_{3} provided that g_{2}^{3} is not equal to 27g_{3}^{2}.
2. Such a lattice is unique.
Best,
S. R. Gal
© 2006 John Baez
baez@math.removethis.ucr.andthis.edu
