# A Journey Through the Mathematical Theory of Braids

A Journey Through the Mathematical Theory of Braidsnis divided into four chapters, each of about 15 minutes. The first contains the basic concepts: the formalization of braids, the group structure on the braids’ set, and the Artin presentation of the braid group. A braid is now described by a word on a set of letters, the generators.

The second chapter deals with the word problem: when do two words represent the same braid? Two algorithms are presented to solve this problem, the Artin braid combing, and the handle reduction.

In the third chapter, knots are presented and put in relation with braids. In the final part, the Jones polynomial is introduced: it is a powerful knot invariant defined through braids.

The last chapter describes braids as dances, that is, motions of points in the disc. The Hilden group, a subgroup of the braid group, is defined and related to closing braids to obtain knots.

### Ali Kaya

This is Ali. Bespectacled and mustachioed father, math blogger, and soccer player. I also do consult for global math and science startups.