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Published online by Cambridge University Press: 06 January 2010
The original motivation for Artin to introduce braids was their potential applications in the study of links. In this chapter we present therefore a few such applications. First we prove that every link can be obtained by closing a braid. This result is implicitely contained in a paper of Alexander from 1923. Then we give an introduction to a highly nontrivial and difficult theorem announced by Markov 1935, but apparently first proved in complete detail by Birman in her book from 1974. Markov's theorem turns the topological problem of classifying link types in euclidean 3–space into an algebraic problem involving the family of Artin braid groups for all numbers of strings. Next we present a proof due to Birman of a presentation theorem stated by Artin 1925 for the group of a link. Finally, we show how to obtain braid representations for links and give an example of computing the corresponding group of a link.
In the last few years there has been a tremendous development in the theory of links not the least due to the discovery of a new polynomial invariant for links made by V.F.R. Jones in 1985. Jones used a representation of the braid group to the group of units in certain Hecke algebras to define his (Laurent) polynomial. The Jones polynomial has by now been generalized to a two variable (Laurent) polynomial and a completely elementary approach to such polynomials has been found. A short guide to the literature and a very pleasing introduction to this subject is given in the following paper by two of the principal investigators:
W.B.R. Lickorish and K.C. Millett: The New Polynomial Invariants of Knots and Links.
Mathematics Magazine 61(1988), 3–23.To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
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