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March 20, 2008

This Week’s Finds in Mathematical Physics (Week 261)

Posted by John Baez

In week261 of This Week’s Finds, learn about the Engraved Hourglass Nebula:

Then read an ode to the number 3, which explains how all these entities are connected:

  • the trefoil knot
  • cubic polynomials
  • the group of permutations of 3 things
  • the three-strand braid group
  • modular forms and cusp forms

The discussion of modular forms here harkens back to this blog entry. I hope to keep explaining modular forms from more and more viewpoints as I keep learning about them in my coffee-fueled chats with James Dolan. This is just the beginning…

I’m going to Singapore the day after tomorrow. My presence here will be fitful at best until I return to Riverside on Thursday March 27th.

In the meantime, here’s a challenge. What are all the ways the trefoil knot shows up in mathematics? What is the most profoundly significant?



My guess for the most profoundly significant is explained in This Week’s Finds.

And here’s another puzzle I don’t know the answer to. Can one exploit the relation between the knot quandle of the trefoil and the vector cross product in 3 dimensions to do something interesting? This knot quandle has 3 generators i,ji, j and kk and 3 relations:

ij=k i \lhd j = k jk=i j \lhd k = i ki=j k \lhd i = j

When we turn it into a group we get the fundamental group of the trefoil complement, or the 3-strand braid group, as explained This Week.

Posted at March 20, 2008 7:14 AM UTC

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Re: This Week’s Finds in Mathematical Physics (Week 261)

Hi John,

This book on modular forms you posted, http://www.math.lsa.umich.edu/~idolga/modular.pdf, are missing the pictures. I use the free acrobat 7, but since that file is from 2005, i don’t think it is my adobe´s reader fault…

Posted by: Daniel de Franša MTd2 on March 20, 2008 7:32 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

The book is very good even if it’s missing some pictures. Maybe I’ll tell the author about this problem, though.

Posted by: John Baez on March 21, 2008 1:15 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

A trefoil knot can also be created by over twisting in making a Moebius band and then cutting down the center.

Also by cutting a donut after first pierceing it with a knife.

cf. Martin Gardner somewhere in his works

Posted by: jim stasheff on March 20, 2008 8:46 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

Here is one more method to produce a trefoil. This method was discovered by J.-L. Loday. I am sure Jim will like it.

Take an edge of a 3d associahedron which is the boundary of a square. Draw a line from the midpoint of this edge to the midpoint of the opposite edge of the square. This last edge is on the boundary of a pentagon and this pentagon has only one other edge, which is on the boundary of a square. Connect the midpoints of these edges by a line. Continue to do the same. After some steps we will return to the initial square but to an edge which was adjacent to the initial edge. Connect the midpoints of these edges but think that this line overcrosses the line on the square which was drawn before. Do the same thing for other squares alternating overcrossings and undercrossings. The final result will be a trefoil.

Posted by: Michael Batanin on March 21, 2008 1:29 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

Wow! How did you discover this? Does it mean anything?

Posted by: John Baez on March 21, 2008 5:40 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

This is not my discovery. Jean-Louis showed it to me when I was visiting him.
We both would like to know very much what does this mean?

Posted by: Michael Batanin on March 21, 2008 8:09 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

If you put the associahedron on the Riemann sphere with squares at 0,1,oo and omega and omega bar sitting at the poles, then the associahedron looks like a pair of pants made of two halves glued together. Then by drawing the trefoil you see a trivalent ribbon vertex around omega and one around omega bar, so the trefoil gives you the Grothendieck ribbon graph. In M theory, we want new weak n-cat axioms which incorporate these weird self referential aspects of number theory. For example, each crossing can be thought of as a stringy dimension: so by going to the 24 squares of the permutoassociahedron on some kind of categorification process, we get to the 24D of the Leech lattice. What knot do we get here?

Posted by: Kea on March 22, 2008 6:33 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

In M theory, we want new weak n-cat axioms which incorporate these weird self referential aspects of number theory. […]

Marni, what is the point of saying things like that? Are you hoping that we ask you what you mean? Are you happy if we just ignore it?

I’d enjoy having a serious conversation about the buzzwords that appear, but the way they appear makes me less than happy.

Even though it has not really happened yet, I usually like to think of the nn-category café as an online place where it should be possible to have serious, sober, technical, non-circular, worthwhile, rewarding discussion about higher structures in physics, and be it string theory.

Since I know well that you don’t appreciate string theory, I suspect, to be frank, that when you mention words like “M-theory” you are trying to engage in a weird kind of parody.

If that is so, please don’t. It’s time to show the interested lurkers out there that sensible substantial discussion about this stuff online is possible. As it is.

Physicists have inadvertedly come a cross an amazing source of higher mathematical structures whose richness and importance is in stark contrast to the lack of serious mathematical attention they are getting. But the average pure mathematician these days who follows the gossip without knowing where to look for the hard statements is bound to get the impression that there is just weirdness to be expected when the M-word appears. That’s very unfortunate. And I hate to think of the nn-category café as contributing to that.

Posted by: Urs Schreiber on March 22, 2008 11:09 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

Urs, when I say M theory, I mean M theory, but since this term is not well defined, I do not necessarily mean the same thing that you do (I don’t). My background is in experimental physics. If this precludes me from commenting on your n-cat cafe, so be it.

Posted by: Kea on March 23, 2008 6:48 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

Not at all - you are most welcome
but please tell us what M-theory means to you

Posted by: jim stasheff on March 23, 2008 2:21 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

Oh, for some pictures

Kea wrote:
associahedron on the Riemann sphere with squares at 0,1,oo

That I can see

and omega and omega bar

what are those?

sitting at the poles,

what poles?

then the associahedron looks like a pair of pants made of two halves glued together.

you mean cut out disks around 0,1,oo ?

Then by drawing the trefoil you see a trivalent ribbon vertex around omega and one around omega bar, so the trefoil gives you the Grothendieck ribbon graph.

somehow this brought to mind a ribbon graph for upper case theta with the cross bar ribbon twisted appropriately

Posted by: jim stasheff on March 23, 2008 2:20 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

Thank you, Jim. By omega I mean the point on the unit circle at real part 1/2, which is a ‘pole’ simply by virtue of being off the real axis. Yes, get the Theta graph on the pants by drawing polygons (squares here) around 0,1,oo.

Just a couple of remarks: I expect M theory to give us new physical observables via motivic techniques, but I do not expect them to agree with mainstream string theory. It is still technically M theory, however, if string structures are recovered in some domain. We don’t think quantum gravity should define observables in terms of classical spaces (continua) or gauge theory at all (or n-cat analogues like n-connections) for these usually assume a concept of universal observer which is alien to our idea of quantum causality. Rather, the finite constraints of a measurement question for a quantum observer, viewed in terms of diagrammatic logic, should be easily expressible in the formalism chosen. As a guide, quantum computation concretely relies on qubit logic, and Brannen’s mass operators are related to qutrit logic, so we want the number $n$ of meta-truth values to label n-cats at the level of the axioms.

This means rewriting QFT completely, but note that much progress has been made already on this front (eg. Connes, Kreimer, Marcolli). The (let us call it) measurement algebra approach goes back to Schwinger, who never approved of the then emergent (now standard) vacuum concept and discussed instead a ‘false vacuum’. The problem with string theory is that they are searching for a vacuum in a landscape, when the usual vacuum idea is entirely irrelevant to the physical information that we would like to derive from the holography of ‘relative’ observers. Operad techniques are important here because they take exactly this step away from concrete models of ‘space’ to the motivic realm (and this is already well known within mathematical QFT). Sorry if this is a bit confusing.

Posted by: Kea on March 24, 2008 7:32 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

Indeed I do!

Posted by: jim stasheff on March 21, 2008 8:44 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

Your trefoil on torus figure [just above section 5)] reminds me of a Tokamak from fusion power in plasma physics.

In turn, a tokamak is very musch like a Smale solenoid pictured on the front cover of Hasselblatt and Katok ‘A First Course in Dynamics’ which can be viewed on Google books and is discussed on page 333.

Posted by: Doug on March 22, 2008 10:04 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

As one of the folk who built the site on which that trefoil on a torus picture appeared, can I ask if other torus knots appear in any of the other contexts mentioned in this thread?

Posted by: Tim Porter on March 22, 2008 2:37 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 261)

Hi Tim Porter,

Although I suspect that your question is directed to the authors of n-category cafe, I will provide what I think is an intriguing answer.

On wikipedia Alpha Solenoid is an image and text found when seaching on Google for trefoil on solenoid images.
There is an important wiki link to Proteasome which has a very extensive discussion.

I think these are related to the trefoil on torus, since the torus may be abstracted as a solenoid with a ZERO helical angle.

What I find most interesting from a speculative perspective is the possibility that ionic proteins composed of amino acids in a solenoid configuration may allow for electron flow that may generate a weak magnetic field.

The same may apply to nucleic acids in helical solenoid configuration. This may be why helices tend to be associated with information.

The helical solenoid is to electrical engineering as the helical spring is to mechanical engineering.

Posted by: Doug on March 24, 2008 1:20 AM | Permalink | Reply to this

Spaces of lattices

Dear Professor Baez,

This all sounds remarkably similar to the talk I heard by Professor Etienne Ghys at the ICM where he spoke about the dynamical system you get by moving around in the space of lattices in R^2. The slides of this talk are available at
http://www.umpa.ens-lyon.fr/~ghys/articles/icm.pdf
and Professor Terence Tao has an excellent overview of the talk available, at
http://terrytao.wordpress.com/2007/08/03/2006-icm-etienne-ghys-knots-and-dynamics/

Posted by: Anon on March 23, 2008 11:35 PM | Permalink | Reply to this

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