1.15. Braid group The braid group on n strands, denoted by Bn, is a set of operations has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number. Braid groups find applications in knot theory, since any knot may be represented as the closure of certain braids. From the algebraic point of view, the braid group is represented in terms of generators βi, with 1 ≤ i ≤ (n−1); βi is a counterclockwise exchange of the ith and (i +1)th strands. β

−1 i is therefore a clockwise exchange of the ith and (i +1)th strands. The generators βi satisfy the defining relations, called Artin relations (see Figure 1.11).

βi βj = βj βi for | i −j |≥ 2

βi βi+1 βi = βi+1 βi βi+1 for 1 ≤ i ≤ n−1 The second is also called Yang–Baxter equation. The only difference from the permutation group is that β2 i = 1, but this is an enormous difference: while the permutation group is finite (the dimension is n!), the braid group is infinite, even for just two strands.
The irreducible representation of the braid group can be given in terms of g ×g dimensional unitary matrices, βi →γi, where thematrices γi satisfy the Artin relations.

a. Consider the group B3. Prove that γ1 = e−7iπ/10 0 0 −e−3iπ/10 , γ2 = −τ e−iπ/10 −i √
τ
−i √
τ −τ eiπ/10 
provide a representation of the Artin relations. Above τ = (
√
5−1)/2, which satisfies τ 2 +τ = 1.

b. Both matrices γi (i = 1,2) are matrices of SU(2) and can be written as γi = expi θi ni 2 · σ
where σj (j = 1, 2,3) are the Pauli matrices and θi is the angle of rotation around the axis  ni . Identify the angles and the axes of rotation that correspond to γ1 and γ2.

c. By multiplying the γi (and their inverse) in a sequence of L steps, as in the example below AL = γ1 γ2γ
−1 1 γ2 . . . γ1  L 
, one generates another matrix AL of SU(2), identified by the angle α of rotation around an axis n, AL = expi α n 2 · σ. Argue that, by making L sufficiently large, find a string of γi and their inverse that approximates with an arbitrary precision any matrix of SU(2).