Narrow Results

In this paper, we give a bound for the number of boundary slopes of orientable immersed proper π₁-injective surfaces of given genus g in an orientable Haken 3-manifold M with a torus boundary, where the bound is independent of M, and a function of g and m, the number of the Jaco–Shalen–Johannson decomposition tori of M.

We investigate the question of when distinct branched surfaces in the complement of a 2-bridge knot support essential surfaces with identical boundary slopes. We determine all instances in which this occurs and identify an infinite family of knots for which no boundary slopes are repeated.