UC Santa Barbara
Braid Groups and Euclidean Simplices
- Author(s): Chisholm, Elizabeth
- Advisor(s): McCammond, Jon
- et al.
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