Ropes, fractions, and moduli spaces

TL;DR

This paper explains Conway's tangle trick, exploring its mathematical basis through braid groups and elliptic curves, connecting it to classical 19th-century mathematics and providing conceptual insights.

Contribution

It offers a detailed exposition and conceptual explanation of Conway's tangle trick, linking it to elliptic curves and classical mathematical theories.

Findings

The tangle trick is rooted in the relationship between braids and elliptic curves.

Mathematical explanation involves group actions and fundamental groups.

Connections to classical work by Weierstrass, Abel, and Jacobi are established.

Abstract

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.