A foliated disk whose boundary is Morton's irreducible 4-braid

Abstract

From the viewpoint of Birman and Menasco, a particular type of singular foliation on a surface with boundary induces an embedding of the surface in 3-space, such that the boundary of the surface is braided relative to the z-axis; hence the foliation determines a conjugacy class in the braid group. Birman and Hirsch describe an explicit algorithm to find a braid word representing this conjugacy class, given the foliation. A braid word β ∈ B_n is said to be irreducible if it is not conjugate to a braid of the form bσ^±1_n−1, with b ∈ B_n−1. We exhibit a foliation of a disk and show that the corresponding braid word is an irreducible element of B₄. We give an explicit geometric description of the embedding induced by the foliation and describe a particularly nice form of symmetry possessed by this example.