Re: Folded Belt



conesetter wrote:
Two, four, or any even number of turns would leave the two looped ends
at the same end of the folded-up pack; whereas three turns, or any odd
number, leave the loops at opposite ends where they are more stable.
--

Perhaps my description was not specific enough. There are no "ends".
Imagine the belt stored on a hook. Starting from any point on the belt
and following it round you pass over the hook three times to get back
to the start-point.


Let's imagine a belt as being a cylinder. It has its circular edges,
and some stuff between, the so-called belt stuff. Just to get the
technical language out of the way.

An uncoiled belt has the property that you could put a soap film
on the top circular edge, and a separate soap film on the lower
edge, without the soap films touching each other or either film
touching the other circular edge. This property is referred to by
saying the boundary circles are *unlinked*.

Now, take a helical arc with an even number of loops, along a (vertical) circular cylinder, and connect the top end to the bottom end by means of some reasonably nice curve. Take a second copy of this arc, translated
by a small distance upwards (small enough so that the spacing between
upper and lower arcs is smaller than the spacing between successive
loops of the helix). Let these two arcs be the upper and lower edges
of a belt.

This second belt will have its two edges *linked* with one another.
I don't know in general what the linking number is (that's a measure
of how many times the two curves wind around each other; there's an
integral formula for it if you must know), but in the case of a
helix having 2 turns in the above description, the curves are linked
once.

As a consequence, when you wind a belt around an even number of turns,
it must acquire a twist in it. It cannot be done while the belt's
belt stuff remains vertical.

There is one caveat: if it were the garment typically called a "belt",
it could be unbuckled, twisted before the winding around, rebuckled,
and you could then form the desired even-loop winding that I've been
pooh-poohing.

Dale

PS I used to believe it was related to the fact that the stable 1-stem
is equal to Z/2Z, but am not so sure any more. I do think it would
be awful swell for stable homotopy to have such a down-to-earth
application, however.
.

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