Also available as http://math.ucr.edu/home/baez/week8.html
This Week's Finds in Mathematical Physics (Week 8)
John Baez
I was delighted to find that Louis Kauffman wants to speak at the
workshop at UCR on knots and quantum gravity; he'll be talking
on "Temperley Lieb recoupling theory and quantum invariants of links and
manifolds". His books
On knots, by Louis H. Kauffman, Princeton, N.J., Princeton University
Press, 1987 (Annals of Mathematics Studies, no. 115)
and more recently
Knots and physics, by Louis H. Kauffman, Teaneck, NJ, World Scientific
Press, 1991 (K & E Series on Knots and Everything, vol. 1)
are a lot of fun to read, and convinced me to turn my energies
towards the intersection of knot theory and mathematical physics. As
you can see by the title of the series he's editing, he is a true
believer the deep significance of knot theory. This was true even
before the Jones polynomial hit the mathematical physics scene, so he
was wellplaced to discover the relationship between the Jones
polynomial (and other new knot invariants) and statistical mechanics,
which seems to be what won him his fame. He is now the editor of a
journal, "Journal of knot theory and its ramifications."
He sent me a packet of articles and preprints which I will briefly
discuss. If you read *any* of the stuff below, *please* read the
delightful reformulation of the 4color theorem in terms of cross
products that he discovered! I am strongly tempted to assign it to my
linear algebra class for homework....
1) Map coloring and the vector cross product, by Louis Kauffman, J. Comb.
Theory B, 48 (1990) 45.
Map coloring, 1deformed spin networks, and TuraevViro
invariants for 3manifolds, by Louis Kauffman, Int. Jour. of Mod. Phys.
B, 6 (1992) 1765  1794.
An algebraic approach to the planar colouring problem, by Louis Kauffman
and H. Saleur, in Comm. Math. Phys. 152 (1993), 565590.
As we all know, the usual cross product of vectors in R^3 is not
associative, so the following theorem is slightly interesting:
Theorem: Consider any two bracketings of a product of any finite number
of vectors, e.g.:
L = a x (b x ((c x d) x e) and R = ((a x b) x c) x (d x e)
Let i, j, and k be the usual canonical basis for R^3:
i = (1,0,0) j = (0,1,0) k = (0,0,1).
Then we may assign a,b,c,... values taken from {i,j,k} in such a way
that L = R and both are nonzero.
But what's really interesting is:
MetaTheorem: The above proposition is equivalent to the 4color
theorem. Recall that this theorem says that any map on the plane may be
colored with 4 colors in such a way that no two regions with the same
color share a border (an edge).
What I mean here is that the only way known to prove this Theorem is to
deduce it from the 4color theorem, and conversely, any proof of this
Theorem would easily give a proof of the 4color theorem! As you all
probably know, the 4color theorem was a difficult conjecture that
resisted proof for about a century before succumbing to a computerbased
proof require the consideration of many, many special cases:
Every planar map is four colorable, by K. I. Appel and W. Haken, Bull.
Amer. Math. Soc. 82 (1976) 711.
So the Theorem above may be regarded as a *profoundly* subtle result
about the "associativity" of the cross product!
Of course, I hope you all rush out now and find out how this Theorem is
equivalent to the 4color theorem. For starters, let me note that it
uses a result of Tait: first, to prove the 4color theorem it's enough
to prove it for maps where only 3 countries meet at each vertex (since
one can stick in a little new country at each vertex), and second,
4coloring such a map is equivalent to coloring the *edges* with 3
colors in such a way that each vertex has edges of all 3 colors
adjoining it. The 3 colors correspond to i, j, and k!
Kauffman and Saleur (the latter a physicist) come up with another algebraic
formulation of the 4color theorem in terms of the TemperleyLieb
algebra. The TemperleyLieb algebra TL_n is a cute algebra with
generators e_1, ..., e_{n1} and relations that depend on a constant d
called the "loop value":
e_i^2 = de_i
e_i e_{i+1} e_i = e_i
e_i e_{i1} e_i = e_i
e_i e_j = e_j e_i for ij > 1.
The point of it becomes clear if we draw the e_i as tangles on n
strands. Let's take n = 3 to keep life simple. Then e_1 is
\ / 
\/ 

/\ 
/ \ 
while e_2 is
 \ /
 \/

 /\
 / \
In general, e_i "folds over" the ith and (i+1)st strands. Note that if
we square e_i we get a loop  e.g., e_1 squared is
\ / 
\/ 

/\ 
/ \ 
\ / 
\/ 

/\ 
/ \ 
Here we are using the usual product of tangles (see the article "tangles"
in the collection of my expository posts, which can be obtained in a
manner described at the end of this post). Now the rule in
TemperleyLieb land is that we can get rid of a loop if we multiply by
the loop value d; that is, the loop "equals" d. So e_1 squared is just
d times
\ / 
\/ 






/\ 
/ \ 
which  since we are doing topology  is the same as e_1. That's why
e_i^2 = de_i.
The other relations are even more obvious. For example, e_1 e_2 e_1 is
just
\ / 
\/ 

/\ 
/ \ 
 \ /
 \/

 /\
 / \
\ / 
\/ 

/\ 
/ \ 
which, since we are doing topology, is just e_1! Similarly, e_2 e_1 e_2
= e_1, and e_i and e_j commute if they are far enough away to keep from
running into each other.
As an exercise for combinatorists: figure out the dimension of TL_n.
Okay, very cute, one might say, but so what? Well, this algebra was
actually first discovered in statistical mechanics, when Temperley and
Lieb were solving a 2dimensional problem:
Relations between the `percolation' and `coloring' problem and other
graphtheoretical problems associated with tregular planar lattices:
some exact results on the `percolation' problem, by H. N. V. Temperley
and E. H. Lieb, Proc. Roy. Soc. Lond. A 322 (1971), 251  280.
It gained a lot more fame when it appeared as the explanation
for the Jones polynomial invariant of knots  although Jones had been
using it not for knot theory, but in the study of von Neumann algebras,
and the Jones polynomial was just an unexpected spinoff. Its importance
in knot theory comes from the fact that it is a quotient of the group
algebra of the braid group (as explained in "Knots and Physics").
It has also found a lot of other applications; for example, I've used it in
my paper on quantum gravity and the algebra of tangles. So it is nice to
see that there is also a formulation of the 4color theorem in terms of
the TemperleyLieb algebra (which I won't present here).
2) Knots and physics, by Louis Kauffman, Proc. Symp. Appl. Math. 45
(1992), 131246.
Spin networks, topology and discrete physics, by Louis Kauffman,
University of Illinois at Chicago preprint.
Vassiliev invariants and the Jones polynomial, by Louis Kauffman,
University of Illinois at Chicago preprint.
Gauss codes and quantum groups, by Louis Kauffman, University of
Illinois at Chicago preprint.
Fermions and link invariants, by Louis Kauffman
and H. Saleur, Yale University preprint YCTPP2191, July 5, 1991.
State models for link polynomials, by Louis Kauffman, L'Enseignement
Mathematique, 36 (1990), 1  37.
The Conway polynomial in R^3 and in thickened surfaces: a new
determinant formulation, by F. Jaeger, Louis Kauffman
and H. Saleur, preprint.
These are a variety of papers on knots, physics and everything.... The
more freewheeling among you might enjoy the comments at the end of the
first paper on "knot epistemology."
I am going to a conference on gravity at UC Santa Barbara on
Friday and Saturday, which I why I am posting this early, and why I have
no time to describe the above papers. I'll talk about my usual
obsessions, and hear what other people are up to, perhaps bringing back
some words of wisdom for next week's "This Week's Finds".

Previous editions of "This Week's Finds," and other expository posts
on mathematical physics, are available by anonymous ftp from
math.princeton.edu, thanks to Francis Fung. They are in the directory
/pub/fycfung/baezpapers. Please don't ask me about hepth and grqc;
instead, read the sci.physics FAQ or the file preprint.info in
/pub/fycfung/baezpapers.