In my post, I made a comment in passing that continued fractions have applications to knot theory. Now A Neighborhood of Infinity has a series of posts on the very topic I had in mind, namely the theory of *rational tangles*. Much of the theory is due to the great John Horton Conway.

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August 26, 2008 at 5:21 pm

John BaezPeople who want to learn about rational tangles may also want to try my quick intro to rational tangles in week228, which has links to these papers:

Jay R. Goldman and Louis H. Kauffman, Rational tangles, Advances in

Applied Mathematics 18 (1997), 300-332. Also available at

http://www.math.uic.edu/~kauffman/RTang.pdf

Louis H. Kauffman and Sofia Lambropoulou, On the classification of

rational tangles, available as math.GT/0311499.

Later I learned a lot more, and wrote about rational tangles, the three-strand braid group and elliptic curves in week233.

Todd has probably talked with Jim Dolan about generalizations of these ideas involving braid groups with more strands, and hyperelliptic curves. I would like to learn more about these and write about ’em sometime.