On the Linear Representation of Braid Groups

Abstract

Abstract.We give a new derivative of the Burau and Gassner representations of the braid and pure braid groups. Various applications are explored.

Monodromy of cyclic coverings of the projective line

“…[1] At the beginning of this subsection, we have verified that the conditions of Lemma 7 are satisfied. By the result of [12] quoted earlier, Ω n admits a nondegenerate hermitian form. Therefore, by Lemma 7, the representation g n (X) is absolutely irreducible.…”

“…We first construct (in subsection 4.1) a skew Hermitian form on the reduced Gassner representation, which is invariant under the action of the pure braid group. The existence of the form is due to [12] (see also [8]) , but the construction for the basis ε i defined in the previous section, may perhaps be new. Using this form, it is easy to decide when specialisations of the Gassner representation (especially d-th roots of unity) are irreducible.…”

“…A (skew) hermitian form preserved by the pure braid group. It is known (see [12] Theorem (3.3)) that the Gassner representation has a skew-hermitian form invariant under the action of the pure braid group. To compute this form with respect to the "ε"-basis {ε i : 1 ≤ i ≤ n} of the reduced Gassner representation, we proceed as follows.…”

The SU(N) Casson–Lin invariants for links

“…Since β = β 1 β 2 is a split braid with β 1 ∈ B k 1 , by (2) we have that β(X) 1 · · · β(X) k 1 = β 1 (X) 1 · · · β 1 (X) k 1 = X 1 · · · X k 1 , and this implies X 1 · · · X k 1 = ω a 1 +···+an 1 X 1 · · · X k 1 . But ω a 1 +···+an 1 = 1 by the assumption of Equation (15), and this gives the desired contradiction.…”

A Casson–Lin type invariant for links

“…, x n . We will follow the conventions of [10] and define the n-string braid group B n to be the subgroup of Aut(F n ) generated by the automorphisms σ 1 , . .…”