Knot Magic Not Magic

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!

---

Muse, March 1999, p. 26-27.

Image credits
Second from bottom and bottom: Knotted strands of DNA. Nicholas R. Cozzarelli, University of California, Berkeley, and Steven A. Wasserman, University of Texas at Austin