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This book studies diverse aspects of braid representations via knots and links. Complete classification results are illustrated for several properties through Xu´s normal 3-braid form and the Hecke algebra representation theory of link polynomials developed by Jones. Topological link types are identified within closures of 3-braids which have a given Alexander or Jones polynomial. Further classifications of knots and links arising by the closure of 3-braids are given, and new results about 4-braids are part of the work. Written with knot theorists, topologists,and graduate students in mind, this book features the identification and analysis of effective techniques for diagrammatic examples with unexpected properties.

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1. Introduction.- 2. Preliminaries, basic definitions and conventions.- 3. Xu´s form and Seifert surfaces.- 4. Polynomial invariants.- 5. Positivity of 3-braid links.- 6. Studying alternating links by braid index.- 7. Applications of the representation theory.- Appendix. -References.-Index.

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"This book contains various interesting and detailed properties of polynomial invariants of closed 3-braids (or 4-braids). This makes a nice complement to a survey by J. S. Birman and W. W. Menasco ... where properties of closed 3-braids, mainly focused on the classification theorem, are summarized." (Tetsuya Ito, Mathematical Reviews, August, 2018)

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