Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Wednesday, May 14, 2008

M Theory Lesson 188

The 1997 Broadhurst and Kreimer paper shows how knot crossing numbers correspond to the weight of the MZV. For example, the positive braid in $B_{2}$ defined by the word $\sigma_{1}^{5}$ is decorated with three chords, and this corresponds to $\zeta (5)$ at weight $w = 5$. The trefoil knot $\sigma_{1}^{3}$ is the simplest $B_{2}$ knot, which gives a three loop chord diagram (well, Feynman diagram, actually) using only two chords. The pattern of crossing and non-crossing chords gets more interesting for braids with $s$ strands where $s > 2$, via the relations of the MZV algebra, where depth corresponds to $(s - 1)$ (this is why the example of $\zeta (5)$ only has one argument). Who would have thought it was so easy to do QED with knots and number theory? Once upon a time physicists admitted that group and gauge theory was a complicated, messy business, so why bother with it? M Theory is much more fun. Observe that the number of points on the circle of the chord diagram is $2n$ (or $w + 1$) where $n$ is the number of chords, so $\zeta (5)$ is really a decorated hexagon, our favourite polygon, often used to label the vertices of the three dimensional associahedron.

14 Comments:

Anonymous a quantum diaries survivor said...

Hi Marni,

just wanted to let you know that I mentioned you and your blog in my video (in italian!!!) on scientific divulgation, for the sci.bzaar.net event I discussed a few times in my blog recently.

You were taken as an example of how one can make scientific divulgation while keeping intact the content, and of a very particular style of blogging - you have a book online by now!

Cheers,
T.

May 15, 2008 6:17 AM  
Blogger Kea said...

Oh, Tommaso, that's naughty of you! But thank you very much. Yes, I would like to write a book, but as I have said previously, I will first need some sort of stable existence.

May 15, 2008 8:51 AM  
Anonymous Anonymous said...

Jeez, what a word, "divulgation"! I just had to Google it because I've never heard it before. It's synonymous with "publication", silly me. Kea, if you're really desperate for stability, get married!

May 15, 2008 10:02 AM  
Blogger Kea said...

Dear anonymous, and how would getting married (to some random person I don't care about) improve my status as a role model for young women who dream of ripping apart the patriarchal establishment with the new tool of Web II, aka Big Sister?

May 15, 2008 10:25 AM  
Blogger CarlBrannen said...

I think it's best to model the external world rather than to mold oneself into being a model. And Madame Curie did okay despite getting hitched. (And what was her maiden name, anyway?)

May 15, 2008 11:00 AM  
Blogger Kea said...

Ah, Carl, but that was then and this is now ....

May 15, 2008 11:02 AM  
Blogger CarlBrannen said...

Kea, well you're not alone. I've got my buddy's TV on while I use his internet to put together my lecture for next week's APSNW 2008 meeting. The program right now is Ax Men, which is about loggers in the Pacific Northwest.

The show reminds me of my job (which is also outdoors in the NW). Heavy equipment. Risk of death from various things. Bunch of rough guys whose language is so bad that the audio is a series of bleeps. Working in really bad weather, cold or rainy. Equipment getting stolen.

At least we've never had anyone severely injured or killed on the job. But I know how it happens.

And along this line, Tesla dug ditches in New York City after the market crash in 1929 (he was 73 years old). He had also worked construction back just after his fued with Edison started. By the way, my somewhat infamous great grandfather was a buddy of Tesla.

So soldier on. It's a rough planet. But history has been rather kind to Tesla.

May 15, 2008 5:58 PM  
Blogger Kea said...

Oh, I'm not complaining, Carl. Your Tesla connection is cool. I quite liked the portrayal of him in the recent movie The Prestige.

May 15, 2008 6:15 PM  
Blogger CarlBrannen said...

I saw Prestige, great movie.

When Edison lost the "current wars" over DC versus AC, a bunch of equipment went surplus and my ggf Elliott bought a generator to provide his house with electricity. He also built his own refrigerator before the commercial models were available.

He hooked up a motor to rotate the family Christmas tree. This annoyed his wife because it involved cutting a hole in the floor.

Back then, I would guess that most electrical engineers had knowledge that now is concentrated in just a small number, such as how to wind motors.

May 16, 2008 10:57 AM  
Blogger Kea said...

Carl, that's an amazing family history you have! The only ggf of mine that I know anything about spent most of his life in prison for attempting to murder the whole family (my grandfather included). That was a WWI incident, related to the difficulty of being German in the South Pacific at the time.

May 16, 2008 11:53 AM  
Blogger CarlBrannen said...

Kea, I have a few of those too, LOL. I'm going to structure my speech tomorrow at the APS meeting around solutions for QCD bound states.

May 17, 2008 10:16 AM  
Blogger L. Riofrio said...

Fascinating history you have, carl. For Kea: In Australia having convict ancestors is a matter of pride! Tasmania has a tour called "Louisa's Walk" following the footsteps of woman prisoners.

May 18, 2008 2:32 AM  
Blogger Kea said...

Hi Louise! Yes, in fact I am also descended from early Australian convicts.

May 18, 2008 6:25 AM  
Blogger CarlBrannen said...

I recently read a most fascinating description of a few, I think Tasmanian, convicts who pirated a ship and sailed it to South America. A nasty life back then.

Hey, they're making the "word verification" things harder and harder.

May 19, 2008 1:27 PM  

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