Faithfulness Properties of the Burau Representation
Date of Award
2005
Document Type
Thesis
Degree Name
Bachelors
Department
Natural Sciences
First Advisor
Mullins, David
Keywords
Burau Representation, Braid Group/Faithful
Area of Concentration
Mathematics
Abstract
This paper is intended as a unified treatment of the existing results on the faithfulness of the Burau representation. Such a treatment is desirable because the articles cited often use conflicting conventions, sometimes even within a single paper. We present the results in terms of a single set of conventions, and we clarify or clean up proofs as needed. We introduce the n-strand braid group Bn as a set of homotopy classes of maps on Dn. We then define a covering space Dn of Dn and note that as we defined it, Bn acts naturally on the relative homology of Dn . We use this action to construct the Burau representation of Bn. Switching to a reduced form of the representation allows us to prove the Burau representation is faithful for n < 3. The representation is shown to be unfaithful for n > 5 by means of a curve in D5 with certain homological properties.
Recommended Citation
Hussey, Caleb, "Faithfulness Properties of the Burau Representation" (2005). Theses & ETDs. 3537.
https://digitalcommons.ncf.edu/theses_etds/3537
Rights
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