A Topological Menagerie

Abstract

Some of the party games that young teenagers play have surprisingly rich mathemati cal content. Entanglement is one such game in which couples are linked with strings tied tightly to their wrists, as shown in Figure la, and then challenged to disentangle themselves, often leading to Figure lb. In fact there is a quick solution (needless to say missing the point of the game): simply push a bit of the girl's string under the boy's at his wrist, pass the resulting loop over his hand, and then pull it free on the other side, as shown in Figure 2. One way to formulate this mathematically is to ask for an isotopy to remove the meridian loop m from a rigid wire w embedded in the 3-sphere, as shown in Figure 3a. (The wire represents the boy with his attached string, where the upper loop of the wire is his right hand, while the meridian represents the girl with her string.) Of course this is easy: just slide m along w to a point just below the upper loop of w, and then stretch it out and pull it off the top. Or put differently, observe that w is isotopic to the trivial wire o-o and so the meridian slips right off.