Braids

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Braids

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The film was realized as part of the Ph.D. thesis of the author. Music: Raul Masu.
Ester Dalvit (Department of Mathematics, University of Trento)
Supported by: 
Département de mathématiques, Université de Trento Département de mathématiques, Université de Caen Matematita, Centre de recherches inter-universitaires pour la communication et l'apprentissage informel des mathématiques.

A journey through the mathematical theory of braids.

The movie is divided into four chapters, each of about 15 minutes.

The first contains the basic concepts: the formalization of braids, the group structure on the set of braids and 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 that was 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 another way of closing braids to obtain knots.

 

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