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UBC Theses and Dissertations

Braid groups, orderings, and algorithms Kim, Djun Maximillian

Abstract

We define a natural and readily calculated bi-invariant strict total ordering of the n-strand pure braid group Pn. It is defined algebraically, using the Artin decomposition of Pn as a semi-direct product of free groups, together with a specific ordering of free groups using the Magnus expansion. This definition extends to define bi-orderings of more general semi-direct products involving free groups, including the fundamental groups of the complements of fibretype hyperplane arrangements. After basic properties of this "Artin-Magnus" ordering are established, we compare it to existing orderings on Pn, including the restriction of the Dehornoy ordering to Pn. Finally, we present algorithms to compute the Dehornoy ordering on the full braid group Bn, and the Artin-Magnus ordering on Pn. An implementation of these algorithms is included as part of a library of objects for symbolic computation with braids.

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