III.4 Braid Groups by F. E. A. Johnson: Page 161.

A braid determines a permutation by the rule the i th hole on the first plane z  label of the corresponding i th hole on the second plane.  This is a surjective homomorphism Bn z Sn that maps si to the transposition (i, i+1).  This is not an isomorphism because Bn is infinite.  si has infinite order and the transposition (i, i+1) squares to the identity (and is finite).

In his 1925 paper “Theorie der Zöpfe ” Artin showed that multiplication in Bn is completely described by the relations:

sisj = sjsi  (|i-j| 2),

sisi+1si = si+1sisi+1.

In physics the Artin relations are called the Yang-Baxter equations.  They are important in  statistical physics.

Artin also devised the method of “Combing the Braid” which allowed for the identification of the identity element from an arbitrary word in the generators.  An alternative algebraic method by Garside (1967) also decides when two elements are conjugate.

In 2001 a proof by Bigelow and independently by Kramer proved that braid groups are linear.

The braid groups described here are braid groups of the plane, because we started with the two parallel planes being punctured.  This method may be generalized to more general curves than the plane.

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