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On the image of the Lawrence-Krammer representation. (English) Zbl 1079.20052
Summary: 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 independent unit complex numbers, then the form is negative-definite Hermitian. Using the fact that non-invertible knots exist this result implies 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 as braids. The two primary tools involved in constructing the sesquilinear form are Bigelow’s interpretation of the Lawrence-Krammer representation, together with the Morse theory of functions on manifolds with corners.
MSC:
20F36Braid groups; Artin groups
20C15Ordinary representations and characters of groups
57M07Topological methods in group theory