Mr James Foley

Honours thesis

Weaving through the braids

The group of n-strand braids, Bn , is defined in various ways: physically as either a collection of tangled up strings or as particles moving about in some space, and algebraically as a presentation of generators and relations. Some algebraic topology is developed, and it is shown that the braid group can also be considered as the fundamental group of a particular configuration space. This allows a more general definition of braids, and presentations are given for braids not only on planes, but also on the torus, sphere, and real projective plane. It is noted that braid theory is trivial on manifolds of dimension greater than two. We then take a different tack, moving from topology to algebra by considering representations of the braid groups. Particularly, we examine the Burau representation and its properties, and then study the representations of an important subgroup of Bn , the symmetric group ?n . This leads to the Lawrence-Krammer representation of braids, which is linear. Although the focus is algebraic, many of the results have been shown via topological methods. We then investigate the connections between knots and braids, show that all braids can be expressed as knots, and that the converse is also true. The Jones Polynomial is established as a knot invariant, and its derivation from braids is demonstrated. Finally, the cryptological possibilities of braid theory are explored, and two particular models, the commutator protocol and the Diffie-Hellman conjugacy protocol, are presented. These are shown to be insecure, and to conclude we brie?y consider some of the major problems to be solved in this area.