Bangert, Patrick David; (2002) Algorithmic problems in the braid groups. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

We introduce the braid groups in their connection to knot theory and investigate several of their properties. Based on term rewriting systems, which we review, we find new solutions to the word and conjugacy problems in the braid groups. A similar problem asks for the minimal length word for an equivalence class in a given braid group which we prove to be NP-complete (after a review of this concept) and present a new algorithm for it. As this algorithm takes an exponentially increasing amount of time, we construct an algebraic approximation algorithm which we find to work well. We consider several methods of approximating the minimal word via computer simulation of the braid strings moving under the influence of certain forces. Using the theory of tangles which we also review, we construct a new notation for knots which is usable by a computer. From this notation, we construct an efficient algorithm to find the braid or plat whose closure is ambient isotopic to any given knot. Finally, we apply the computer software developed for these problems to the solar coronal heating problem by simulating magnetic flux tubes. We also present a number of incidental results that were found along the way of researching these problems.

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