  A non-singular sesquilinear form is constructed that is preserved by the
Lawrence-Krammer representation. It is shown that if the polynomial variables q
and t of the Lawrence-Krammer representation are chosen to be appropriate
algebraically independant unit complex numbers, then the form is
negative-definite Hermitian. Since unitary matrices diagonalize, the conjugacy
class of a matrix in the unitary group is determined by its eigenvalues. It is
shown that the eigenvalues of a Lawrence-Krammer matrix satisfy some symmetry
relations. Using the fact that non-invertible knots exist, the symmetry
relations imply that there are matrices in the image of the Lawrence-Krammer
representation that are conjugate in the unitary group, yet the braids that
they correspond to are not conjugate. The two primary tools involved in
constructing the form are Bigelow's interpretation of the Lawrence-Krammer
representation, together with Morse theory on manifolds with corners.
