On the image of the Lawrence-Krammer representation

Abstract

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.