| In this thesis, various ideas relating to algebras with homomorphic braid group images are studied. An alternate basis is found for the vector space underlying the Temperley-Lieb algebra, which relates to the braid group presentation used by Birman, Ko, and Lee to solve the word and conjugacy problems. In the Birman-Murakami-Wenzl algebra, two irreducible representations are studied, and are found to be equivalent (up to one-dimensional tensor) to the faithful Krammer-Lawrence representation, proving that the algebra is faithful and the representation is irreducible.; This representation is further investigated for ways to read information about a braid from its matrix. A correlation is found between Bigelow's intersection pairing of forks and noodles, and the column vectors of matrices in the Krammer-Lawrence representation. It is established that the information in the matrices is sufficient to detect and distinguish Lorenz braids, horseshoe braids, and fixed strands. |
