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MatheMUSEments
Knot Magic Not Magic
By Ivars Peterson
Muse, March
1999, p. 26-27.
Have you ever watched a magician
tie a humongous knot
and
then, as if by magic, make it fall apart? Sometimes what looks
like an impressive knot isn't a knot at all. Magicians and escape
artists are experts at tying phony knots.
Here's an example you can use to
impress your
friends. Get a
piece of rope—a few feet long—and follow these steps.
A tug should easily undo the
resulting tangle.
Though magicians like knots that
can fool people,
mathematicians are mostly interested in real knots, especially
knots that can't be undone. A mathematician's knot is always in a
piece of rope that has both ends stuck together. Without cutting
the rope, there's no way you can get the knot out.
You can make a mathematician's knot
by plugging an
electrical
extension cord into itself and making a single loop. That's
called an unknot.
But if you cross and wind the cord
around itself a
few times
before plugging it in, you'll likely end up with a true knot. How
many different knots can you make? To tell one from another, you
can start putting them into groups by counting how many times the
cord crosses itself when you lay each knotted loop down on a
table. The simplest possible knot, called a trefoil, has three
crossings.
Mathematicians started to describe
all the different
kinds of
knots more than 100 years ago. They now know there are exactly
1,701,936
knots with 16 or fewer crossings. (Imagine trying to draw
pictures of all of them!) Along the way, however, they were
fooled again and again. Sometimes two knots looked different, but
were really the same. Other times, something that looked like a
knot was really an unknot. To keep from getting fooled,
mathematicians looked for formulas that would serve as shortcuts
for telling a knot from an unknot and one knot from another.
They're still looking for a single formular that covers all
possible knots.
Most mathematicians study knots
just because they
enjoy it.
But figuring out knots can help others understand more about how
the world works. Biologists, for example, study knots so tiny you
need an electron microscope to see them. Heaps of DNA strands sit
like microscopic spaghetti inside plant and animal cells. DNA
carries the code that tells each cell what to do.When scientists
started to sort out the DNA strands found in cells, they
discovered some unexpected loops and knots.
Mathematicians came to the rescue,
showing how it was
possible
to snip a DNA strand in two, then rejoin several split strands to
make each of the loops and knots the scientists had seen. That
tells biologists a lot about the way cells manipulate and
duplicate DNA. They end up with a better understanding of how
drugs, viruses, and other things can alter DNA's elaborate
tangles.
There's a lot more to knots than
tying your
shoelaces!
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