SOME LINEAR REPRESENTATIONS OF BRAID GROUPS

In this paper we exhibit a way of obtaining linear representations of the braid groups B_n over ℤ[t] by studying their action on the set of isotopy classes of sets of simple closed curves on a punctured disc. The cases n=3, 4, 5 are shown to be very different from the cases n>5. We show a connection between representations of B₃ and Pascal's triangle. We also show that there is a sequence of polynomials κ_i(t), i≥0, related to polynomials P_i(t) defined by V. F. R. Jones all of whose roots give values of t for which these representations are not faithful.