Ropes, fractions, and moduli spaces

This is an exposition of John H. Conway's tangle trick. We discuss what the trick is, how to perform it, why it works mathematically, and finally offer a conceptual explanation for why a trick like this should exist in the first place. The mathematical centerpiece is the relationship between braids on three strands and elliptic curves, and we a draw a line from the tangle trick back to work of Weierstrass, Abel, and Jacobi in the 19th century. For the most part we assume only a familiarity with the language of group actions, but some prior exposure to the fundamental group is beneficial in places.Comment: Expository article. 13 pages with 10 figures. Suitable for graduate students and advanced undergrads. Comments welcome! New in V2: error in Example 2.1 corrected, brief discussion of variants of the trick, acknowledgements adde