Toroidal and Klein bottle boundary slopes

Abstract

Let M be a compact, connected, orientable, irreducible 3-manifold and T 0 an incompressible torus boundary component of M such that the pair (M, T 0 ) is not cabled. In the paper "Toroidal and Klein bottle boundary slopes" [5] by the author it was established that for any, the maximal number of mutually parallel, consecutive, negative edges that may appear in G Fi is n j + 1, where n j = |∂F j |. In this paper we show that the correct such bound is n j + 2, give a partial classification of the pairs (M, T 0 ) where the bound n j + 2 is reached, and show that if ∆(∂F 1 , ∂F 2 ) ≥ 6 then the bound n j + 2 cannot be reached; this latter fact allows for the short proof of the classification of the pairs (M, T 0 ) with M a hyperbolic 3-manifold and ∆(∂F 1 , ∂F 2 ) ≥ 6 to work without change as outlined in [5].