From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Tangles
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Thu, 8 Oct 92 03:46:04 GMT


Before I finally get around to revealing what a 2-category is and what
they might have to do with quantum gravity, I thought I should talk a
bit about the category of tangles, because it is so utterly beautiful.

I'm not going to be very formal, so anyone who wants the rigorous
details should take a look at 

Yetter D N 1988 Markov algebras, in Braids, Contemp. Math. 78, 705.

Turaev V G 1990 Operator invariants of tangles, and
R-matrices, Math.\ USSR Izvestia 35 411.

I will just say that a tangle is a bunch of strands connecting
n points on the ceiling to m points on the floor, possibly with a bunch
of knots thrown in the middle:



 	|   |
        \   /
         \ /
          \     /\
         / \   /  \
        /   \ /    \
       /     \      \
      /     / \      |
      \    /   \     |
       |  /     \ /\ /
      /   \      \  \
     /     |    / \/ \
    /      |    \____/
   |       |
   |       |

For us two tangles will be the same (technically, "isotopic") if one
can be deformed into the other; i.e., we think of the strands as being
infinitely flexible and are allowed to wiggle them around but not move
them over the ceiling or under the floor; we aren't allowed to move
the places where the strands touch the ceiling and floor, though.

Note that knots, links and braids are all special cases of tangles.
Tangles are great because the provide a nice algebraic structure to
study all of these things.

We say the above tangle is in Hom(2,2) because there are 2 points on the
ceiling and 2 on the floor.  Here is an element of Hom(2,4): 

 	|   |
        \   /
         \ /
          \     /\
         / \   /  \
        |   \ /    \
        |    \      |
        |   / \     |

and here is an element of Hom(4,0):

        |  |   |    |
        \  /   \ /\ /
         \/     \  \
               / \/ \
               \____/

Note that we can "compose" these tangles to get one in Hom(2,0):

 	|   |
        \   /
         \ /
          \     /\
         / \   /  \
        |   \ /    \
        |    \      |
        |   / \     |
        |  |   |    |
        \  /   \ /\ /
         \/     \  \
               / \/ \
               \____/

In Hom(n,n) there is an "identity" tangle which is just a bunch of
vertical strands:

      |   |   |
      |   |   |
      |   |   |
      |   |   |
      |   |   |
      |   |   |

and if you compose any tangle x with the identity on the right or left you
get x again.  This, together with the associativity of composition, is
all we mean by saying that tangles form a category.

But we can also take the tensor product of two tangles.  The tensor
product of 

 	|   |
        \   /
         \ /
          \     /\
         / \   /  \
        |   \ /    \
        |    \      |
        |   / \     |

and

        |  |   |    |
        \  /   \ /\ /
         \/     \  \
               / \/ \
               \____/

is

 	|   |                   |  |   |    |
        \   /                   \  /   \ /\ /
         \ /                     \/     \  \
          \     /\                     / \/ \ 
         / \   /  \                    \____/
        |   \ /    \      
        |    \      |    
        |   / \     |    

Notice that the tensor product is associative and that composition
and tensor product satisfy certain obvious identities (just like the
identities that hold for tensor product and composition of linear maps
between vector spaces).  

By now the physicists must be wondering how mathematicians get paid to
play around with this sort of thing.  I will try to head off such rude
remarks by noting that another name for tangles would be "Feynman
diagrams".  Of course Feynman diagrams are *labeled* tangles - the strands
carry spin, momentum, and other quantum numbers.  Also Feynman diagrams
have vertices, which tangles don't.  Also Feynman diagrams don't really
live in 3-dimensional space.  Nonetheless, there is a real relationship.
For starters, if we were going to talk about labeled tangles with
vertices we would be working with a generalization that has been studied
by Reshetikhin and Turaev in their paper:

Reshetikhin N,  Turaev V 1990 Ribbon
graphs and their invariants derived from quantum groups, 
Comm. Math. Phys. 127, 1.

Their strands are labeled by representations of quantum groups and they
get nice topological invariants this way.  

Anyway, here's the question of the day: how do we describe the category
of tangles.  Well, Turaev and Yetter both showed that it can be
described by generators and relations almost like a group can.  The
generators are as follows.  First, the identity 1 in Hom(1,1):

   |     
   | 
   |     
   |
   |

Second, the basic "right-handed crossing" r in Hom(2,2):

 \   /
  \ /
   /
  / \
 /   \

and the left-handed crossing r^{-1} in Hom(2,2):
 
 \   /
  \ /
   \
  / \
 /   \

(Note that rr^{-1} and r^{-1}r are both the identity in Hom(2,2), so
the names are appropriate.)

Third, the "cup" in Hom(2,0):

 \      /
  \    /
   \  /
    \/

and the "cap" in Hom(0,2):

    /\
   /  \
  /    \
 /      \

These are really primordial things!  In what sense do they "generate"
the category of tangles.  Well, any tangle can be formed from these guys
by taking tensor products and composites.  For example, this guy

 	|   |
        \   /
         \ /
          \     /\
         / \   /  \
        |   \ /    \
        |    /      |
        |   / \     |

can be written as

     (1 x r x 1)(r^{-1} x cap)

where I'm using x for tensor product and juxtaposition for composition.

Okay, so those are the generators (if you don't believe me, prove it!).
What are the relations?  This is the cool part.  Of course, there are a
bunch of relations which just come from the properties of the tensor
product and composition.  Tensor product and composition satisfy these
relations in any "monoidal category", but what we want are the relations
special to the category of tangles.  They were figured out by Turaev and
Yetter, who actually came up with slightly different, but equivalent,
sets of relations.  As I like them, they are as follows... I'll draw
them rather than write them as formulae:


 \   /       \   /       |    |
  \ /         \ /        |    |
   /           \         |    |
  / \         / \        |    |
 /   \    =  /   \    =  |    |
 \   /       \   /       |    |
  \ /         \ /        |    |
   \           /         |    |
  / \         / \        |    |
 /   \       /   \       |    |

(which just says that r^{-1} really lives up to its name, and is also
known as the 2nd Reidemeister move)


\   /    |	|     \ /
 \ /     |	|      \  
  \      |	|     / \
 / \     |	|    /   \
/   \    /	\   /    |
|     \ / 	 \ /     |
|      \     =	  \      |	
|     / \          \     |
|    /	 \	 /  \   /
\   /    |	|    \ / 
 \ /     |	|     \  
  \      |  	|    / \ 
 / \     |	|   /   \
/   \    |	|  /     \

(which is the 3rd Reidemeister move, also known as the Yang-Baxter
equation)

|       /\           /\       |       |
|      /  \         /  \      |       |
|     /    \       /    \     |       |
\    /      |  =  |      \    /  =    | 
 \  /       |     |       \  /        |
  \/        |     |        \/         |


   |           |     /\          |     /\
   |           |    /  \         |    /  \
   |           \   /    \        \   /    \
   |            \ /     |         \ /     |
   |    =        \      |    =     /      |
   |            / \     |         / \     |
   |           /   \    /        /   \    /
   |           |    \  /         |    \  / 
   |           |     \/          |     \/
   |           |                 |      

(which is called the 1st Reidemeister move)

and then 4 closely related identities:

|     \   /      \   /     |
|      \ /        \ /      |
|       \          /       |
|      / \        / \      |
|     /   \  =   /   \     |
\    /    |     |     \    /    
 \  /     |     |      \  /
  \/      |     |       \/
          

|     \   /      \   /     |
|      \ /        \ /      |
|       /          \       |
|      / \        / \      |
|     /   \  =   /   \     |
\    /    |     |     \    /    
 \  /     |     |      \  /
  \/      |     |       \/


  /\      |     |      /\
 /  \     |     |     /  \
/    \    |     |    /    \
|     \   /     \   /     |
|      \ /       \ /      |
|       /    =    \       |
|      / \       / \      |
|     /   \     /   \     |


  /\      |     |      /\
 /  \     |     |     /  \
/    \    |     |    /    \
|     \   /     \   /     |
|      \ /       \ /      |
|       \    =    /       |
|      / \       / \      |
|     /   \     /   \     |


If you want to see these identities written algebraically, and you have
LaTeX, give this a try:

\def\tensor{\otimes}
\begin{array}  rr^{-1} &=& r^{-1}r \quad =\quad | \tensor | \nonumber\cr
  (| \tensor \cup)(\cap \tensor |) &=& (\cup \tensor |)(| \tensor
\cap) \quad =\quad |  \nonumber\cr
  (r \tensor |)(| \tensor r)(r\tensor |) &=& (| \tensor r)(r \tensor
|)(1 \tensor r) \nonumber\cr
(| \tensor \cup)(r \tensor |) &=& (\cup \tensor |)(| \tensor r^{-1}) 
\nonumber\cr 
(| \tensor \cup)(r^{-1} \tensor |) &=& (\cup \tensor |)(| \tensor r)
\nonumber\cr 
(r \tensor |)(| \tensor \cap) &=& (| \tensor r^{-1})(\cap \tensor |)
\nonumber\cr
(r^{-1} \tensor |)(| \tensor \cap) &=& (| \tensor r)(\cap \tensor |)
\nonumber\cr
(| \tensor \cup)(r \tensor |)(| \tensor \cap) &=& (| \tensor \cup)
(r^{-1} \tensor |)(| \tensor \cap = | \tensor |.\nonumber\end{array}

To see the power of these identities (together with the rules satisfied
by tensor product and composition), use them to deduce:

           |     /\            /\     |
           |    /  \          /  \    |  
           \   /    \        /    \   /    
            \ /     |       |      \ /     
             \      |    =  |       \      
            / \     |       |      / \     
           /   \    /        \    /   \    
           |    \  /          \  /    |  
           |     \/            \/     |   
           |                          |

There are endless fun and games to be had with these rules, which
encode a lot of the topology of 3-dimensional space.  Try Kauffman's
book "Knots and Physics" for some of this fun.  It may be surprising,
but it shouldn't be, that the category of tangles and its
representations constitutes a big hunk of conformal field theory (hence
string theory).  It also is practically the SAME THING as Chern-Simons
field theory (a 3-dimensional topological quantum field theory) and, I
attempt to show in my paper "Quantum Gravity and the Algebra of
Tangles," they have a lot to do with 4-dimensional quantum gravity.
Newsgroups: sci.physics