Vanishing triple products.

Abstract

In studying Knot Theory, one fundamental problem is to determine whether two links are equivalent. Many polyomials are defined aximoatically or algebraically which answered partially the question of determining the equivalence of two links. using the Linking Number, one can classify 2-component links into two classes: those that have Linking Number zero and those that do not. Using Triple Products one can classify links with 2-components into two classes: those that have Linking Number zero and those that do not. Using Triple Products one can classify links with 3-components into two classes: those that have all Triple Products zero and those that have at least one non-zero Triple Product. Determining Vanishing Triple Products using the definition is beyond the scope of this thesis since it requires and intensive study of Cohomology Group Theory and Lie Algebras. In this thesis, an algorithm developed by Dr. Stefanos Gialamas is used, in order to detect vanishing Triple Products in the complement of a link with 3-components. The algorithm requires a presentation of the fundamental group of the link (Wirtinger Presenation) and techniques from the Commutator Calculus and the Fox Derivatives. The algorithm is applied to closed braids, which are links, and answers the question which closed braids have all Triple Products vanished?