Conference rusure::math

1540.0. "A braid puzzle." by CAD::COOPER (Topher Cooper) Mon Jan 13 1992 12:24

(NOTE: Also posted to PHYSICS in the Braids and Quantification thread).

From: [email protected] (John C. Baez)
Newsgroups: sci.math,sci.physics
Subject: Re: Braids and Quantization
Message-ID: <[email protected]>
Date: 10 Jan 92 18:23:54 GMT
References: <[email protected]>
Sender: [email protected]
Organization: MIT Department of Mathematics, Cambridge, MA
Lines: 83
Xref: nntpd.lkg.dec.com sci.math:23698 sci.physics:26640


Okay, this is a "just-for-fun" posting on braids before I get back to
the serious stuff.  I had a mind-bending exploration of topology the
other night with a friend that I'd like to recommend to all of you.
Take a long thing piece of paper in it and cut two parallel long slits
in it like so:
   _________________________________________________________________
  |  _____________________________________________________________  |
  |  _____________________________________________________________  |
  |_________________________________________________________________|      


This represents the trivial braid on three strands.  You can actually
get nontrivial braids by "cheating" and grabbing the end marked A
(below) and sliding it through one of the slits --- say the top one,
around the position marked X (the exact position doesn't matter, of
course), and then pulling it back to about its original position:

   _________________________________________________________________
  |  ____________________________________________X________________  |
 A|  _____________________________________________________________  |
  |_________________________________________________________________|      

If you do this in the simplest possible way, you will get a braid in
which two of the strands cross around each other twice, while the third
strand is not tangled with the other two --- but all the strands have a
360 degree twist in them now!!

(So really we are working here not with braids but "framed braids," in
which each strand has a certain twist to it.  Framed braids also form a
group which has the ordinary braid group as a quotient.)

Okay, here's your puzzle.  Get your piece of paper to look like this as
a braid -- with no strand having any twist in it:
 ____   ___   ___   ______
     \ /   \ /   \ / 
      \     \     \
 ____/ \   / \   / \   ___
        \ /   \ /   \ /
         /     /     /
 _______/ \___/ \___/ \___

(What I love about ASCII is its graphics capabilities.)  This sort of braid,
where top and bottom strand take turns going over the middle strand, is
the typical braid found in hairdos.  Here however the exact number of
crossings counts.  Note two neat things about this braid.  First, each
strand winds up in its original position (top to top, middle to middle,
bottom to bottom) - i.e. its image in the symmetric group is the
identity.  Second, if we get rid of any one strand the remaining two are
unlinke (i.e. form a trivial braid on two strands).  Thus it's a braid
analog to the "Borromean rings" (three linked circles no pair of which
are linked).  

Anyway, getting your piece of paper to look like this without any
cutting and pasting is a topological trick well-known to
leather-workers, who can make seamless leather braids this way.  My
friend and I were unable to make this braid except using the following
trick.  Grab the strands near the left (as in the first picture) and
braid them to look like the desired braid, ignoring the fact that near
the right things are getting all screwed up.  Now look at what you have
at the right -- the inverse braid of the one you want (no surprise,
since the whole braid is still the identity braid)!  While preserving the left
half, which is the way you want it, now use "cheating" moves on the
right half (i.e., grab the right end and slip it through the slits) to
kill off the unwanted junk (the inverse braid of the one you want).  You
can do it with three, or perhaps even just two, "cheating moves" --- if
you're clever!  You are now left with the desired braid as in the third
picture!  

Now there has got to be a more straightforward way of doing this!   One
should simply be able to create the desired braid by 2 or 3 cheating
moves.  Unfortunately my friend and I never succeeded.  It's sort of
like we knew how to differentiate but not how to integrate.  But we
learned some interesting topology in the process -- and that's what
counts!  So I strongly recommend that everyone make a 2-slitted strip of
paper (leather would be better) and see what kinds of framed braids they
can make.  There is clearly an interesting sort of group lurking here:
the subgroup of framed braids that can be generated by "cheating moves".
I am sure that topologists have figured this stuff out already, but it's
more fun to mess with it yourself in this case.  If you keep track of
which framed braids you can make by (various types of) "cheating moves"
and try to figure out the pattern, preferably with a friend, you will be
guaranteed hours of mind-boggling fun.

    I sent the following to the author this morning:

---------------------------------------------------------------------------
    Our newsfeed is cut off this morning, so I can't check if, as is
    likely, you've received 999 answers to this, but just in case:

    When I saw this I was pretty sure that I had seen it, but I had to
    check at home.  The answer may be found in Martin Gardner's collection
    of Scientific American columns (where else?), entitled The Unexpected
    Hanging.  The columns in that collection first appeared in the early
    60's.  Specifically, its in the section (column) entitled The Church
    of The Fourth Dimension.  The answer appears on page 74 (your
    "inefficient" answer is also mentioned).

				    Topher