The set of rational tangles has a nice identification with $PSL(2,Z)$. Here's a formal treatment: Rational Tangles and the Modular Group. And here's an easy version aimed at teachers: http://www.geometer.org/mathcircles/tangle.pdf.
Rather than repeat what's written in this papers I'll just say this: We can use the isomorphism, and a bit of continued fractions, to give us a procedure for undoing tangles. I think it gives a wonderful example of how the same abstract group structure appears in two entirely different looking places: topology and (elementary) arithmetic.