Newsgroups: sci.physics.research,sci.math,sci.physics
From: baez@galaxy.ucr.edu (john baez)
Subject: Re: This Week's Finds in Mathematical Physics (Week 125)
Date: Mon, 9 Nov 1998 01:59:40 GMT

Chris Hillman  <hillman@math.washington.edu> wrote:

>I thought I'd suggest a fairly elementary reference for some of this
>stuff.  The advanced undergraduate level textbook by Jones & Silverman,
>Complex Functions, Cambridge U Press, 1987, offers an introduction to the
>hyperbolic space H^2 in the upper half plane model, the Moebius group
>PSL_2(C), the stabilizer PSL_2(R) of the upper half plane, and the modular
>group PSL_2(Z), as well as lattices, elliptic curves, the moduli space,
>the j function, automorphic functions, Fuchsian groups, etc.

Sounds good - I'll go to the library today and check it out if I can.

>While I'm at it, perhaps I can sketch still another beautiful connection,
>between geodesics on the moduli space H^2/PSL_2(Z) and continued fraction
>representations of real numbers.  

Zounds!   It'll take a while to sink in.  But it's clearly related to
John Conway's pleasant game of "rational tangles".  Here two players,
call them A and B, start by facing each other and holding ropes in 
each hand connecting them together like this:
  
A   A
|   |
|   |
B   B
  
This is called "position 0".  The referee then cries out either "add one!" 
or "take the negative reciprocal!".   If the referee says "add one!", player
B has to switch which hand he's using to hold which rope, making sure to
pass the right one over the left, like this:

A   A
 \ /
  /
 / \
B   B

This is called "position 1" since we started with "position 0" and
then did "add one!".  But if the referee says "take the inverse 
reciprocal!" both players must cooperate to move all four ends of
the ropes a quarter-turn clockwise, like this:

A   A
 \_/
  _
 / \
B   B

This is called "position -1/0", since we started with 0 and then
did "take the negative reciprocal!".  

The referee keeps crying "add one!" or "take the negative reciprocal!"
in whatever order she feels like, and players A and B keep doing the
same sort of thing: either player B switches the ropes right over left,
or both players rotate the whole tangle a quarter-turn clockwise.  It's
actually best if the referee doesn't start with "take the negative
reciprocal!", since some people refuse to divide by zero, for religious
reasons.  

Anyway, after a while the ropes are all tangled up and the rope is 
in "position p/q" for some complicated rational number p/q.  Then
the players have to *undo* the tangling and get back to "position 0".
They may not remember the exact sequence of moves that got them into
the mess they are in.  In fact the game is much more fun if they *don't*
remember.  It's best to do it at a party, possibly after a few drinks.

Luckily, any sequence of "add one!" and "take the negative
reciprocal!" moves that carry their number back to 0, will carry 
their tangle back to "position 0".  So they just need to figure out
how to get their number back to 0, and the tangle will automatically
untangle itself.  That's the cool part!

I leave it as a puzzle to the mathematically inclined to figure out how 
this works.  I'm not sure I *completely* understand how it works myself,
but this much is obvious: the move "add one!" corresponds to the matrix 
in SL(2,Z) that everyone calls T, while the move "take the negative 
reciprocal!" corresponds to the matrix everyone calls S.  

>(Note that contrary to "everyone calls" :-/ Jones and Silverman use the 
>notation X for S and Y for ST.)  

Well, okay, so not *everyone* calls them S and T.  But certainly everyone
working in conformal field theory calls them S and T.  They are very
important when studying the 4-point function in conformal field theory, 
but I'd better shut up now, or I'll give away too many clues to the above 
puzzle.   For hints, try Witten's paper "Quantum Field Theory and the 
Jones Polynomial", Comm. Math. Phys. 121 (1989) 351.