The Burau representation of the Braid group

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Ricardo García López[en] This work deals with two relevant subjects of modern mathematics: representation theory and braid theory. It also includes a relation between them; the Burau representation, as well as a new concept; the Fox partial derivatives and its relation with the braid theory, more specifically with the Burau representation. The structure of the work is as follows. Firstly, the basic notions of representation and braid theory will be given to understand the following results. Therefore, the Burau representation will be studied giving its definition, the one of its reduced form and proving its faithfulness. A different interpretation of this representation, introduced by V. Jones will also be studied and it has to be outlined due to its relation with the probability. Finally, the Fox calculus and its partial derivatives will be introduced to show an alternative approach to the Burau representation