\section{Bicycle chains}

Sometimes, when you come to put the rear wheel back on your bike after
fixing a flat, or when you are fooling around trying to get the chain
back onto the sprockets after it has slipped off,
%Fluff.  PGD, January 2006
%(what? you don't always keep your gears adjusted perfectly?)
you may find that the chain is
in the peculiar kinked configuration shown in Figure \ref{kink}.

\fg{kink}{Kinked bicycle chain.}

\begin{problems}
\item
Since you haven't removed a link of the chain or anything like that,
you know it must be possible to get the chain unkinked, but how?
Play around with a bike chain (a pair of
rubber gloves is handy), and figure out how to introduce and
remove kinks of this kind.
\item
Draw a sequence of diagrams showing intermediate stages that you go
through to get from the kinked to the kinked configuration.
%
%Too hard at this point!  PGD, January 2006
%
%\item
%Now close your eyes, and see if you can visualize the process of kinking
%and unkinking the chain.
%This can be tricky at first;
%practice it until you get good at it.
%Visualization is a skill that you can learn.
%The more you practice it, the better you get at it.
%
%Goes without saying.  PGD, January 2006
%
%\item
%Keep a sharp eye out for opportunities to impress people with your
%ability to disentangle bicycle chains.
%Such opportunities are not common,
%but the satisfaction attendant on straightening out someone's
%chain is hard to describe.
\item
Take a look at the bicycle chains shown in Figure \ref{morekinks}.
\fg{morekinks}{More kinked bicycle chains.}
Some of these chain are not in configurations that the chain can get into
from the normal configuration without removing a link.  To disentangle these
recalcitrant chain, you would need to remove one of the links using a
tool called a `chain-puller', mess around with the open-ended chain,
and then do the link back up again.
Can you tell which chains require a chain-puller?
\item
Some of the chains in Figure \ref{morekinks} that require a chain-puller
can be untangled without one if you know how to perform Chain Magic, which
is a magical spell that will convert between
an overcrossing and an undercrossing, as shown in Figure
\ref{chainmagic}.
Which?

\fg{chainmagic}{Chain Magic.}
\item
Try to formulate a general rule that will tell you which chains can
be untangled with Chain Magic, but without the aid of a chain-puller.
\item
Now how about a rule to tell which chains can be untangled without Chain Magic?
\item
The theory of straightening out bicycle chains using Chain Magic is
called `regular homotopy theory'.
A higher-dimensional version of the theory explains how you can turn
a sphere in three dimensional space `inside out'.
What this means and how it is done is explained in the
video `Inside Out', produced by the Minnesota Geometry Center.
Keep your eye out for an opportunity to watch this amazing video.
\end{problems}
