Mathematical knots are a simplified model of everyday knots in the same way that the straight lines of geometry are a mathematical model for an everyday idea. So a knot in mathspeak becomes an infinitely differentiable closed curve in ordinary three dimensional space. Their study began in earnest in the late 19th century and continues today.

Mathematical braids are also a simplified model of everyday braids such as those one ties in one’s hair, in a belt or to make rope. We will discuss both these concepts and show how the braids admit a group structure while knots do not. A simple geometrical operation allows one to obtain knots from braids and thereby take advantage of the group structure to study knots.

Poater (PDF)