The calculus of braids : an introduction, and beyond / Patrick Dehornoy ; translated by Danièle Gibbons, Greg Gibbons.

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Geometric braids -- Braid groups -- Braid monoids -- The greedy normal form -- The Artin representation -- Handle reduction -- The Dynnikov coordinates -- A few avenues of investigation -- Solutions to the exercises.

"Everyone knows what braids are, whether they be made of hair, knitting wool, or electrical cables. However, it is not at all evident that we can construct a theory about them, that is, elaborate a coherent and mathematically interesting corpus of results concerning them. Our goal here is to convince the reader that there is a resoundingly positive response to this question: braids are indeed fascinating objects, with a variety of rich mathematical properties. For this, we will concentrate on carefully and completely establishing only a few selected results. What they have in common is the sophistication of the proofs they require, in spite of their very simple statements. At the heart of the approach, a natural multiplication operation of braids leads to group structures, the braid groups. Combining both algebraic and topological aspects, these groups enjoy multiple interesting properties and are at the same time simple and complex"-- Provided by publisher.

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