UC Santa Barbara

In the early 2000s, Daan Krammer and Stephen Bigelow independently proved that braid groups are linear. They used the Lawrence-Krammer-Bigelow (LKB) representation for generic values of its variables q and t. The $t$ variable is related to the Garside structure of the braid group used in Krammer's algebraic proof. The q variable, associated with the dual Garside structure

of the braid group, has received less attention.

In this dissertation we give a geometric interpretation of the q portion

of the LKB representation in terms of an action of the braid group on

the space of non-degenerate euclidean simplices. In our

interpretation, braid group elements act by systematically reshaping

(and relabeling) euclidean simplices. The reshapings associated to

the simple elements in the dual Garside structure of the braid group

are of an especially elementary type that we call relabeling and

rescaling.