Correction to "Toroidal and Klein bottle boundary slopes"

TL;DR

This paper corrects a previous boundary slope bound in 3-manifold theory, refines the classification of certain manifolds, and establishes that high boundary slope distance prevents reaching the corrected bound.

Contribution

It corrects the previous boundary slope bound, provides a partial classification of manifolds reaching this bound, and links boundary slope distance to the bound's attainability.

Findings

Corrected the maximum number of negative edges to n'+2.

Identified conditions under which the bound is reached.

Proved that a boundary slope distance of at least 6 prevents reaching the bound.

Abstract

Let M be a compact, connected, orientable, irreducible 3-manifold and T an incompressible torus boundary component of M such that the pair (M,T) is not cabled. In the paper "Toroidal and Klein bottle boundary slopes" [arXiv:math/0601034] by the author it was established that for any K-incompressible tori F,F' in (M,T) which intersect in graphs G in F and G' in F', the maximal number of mutually parallel, consecutive, negative edges that may appear in G is n'+1, where n' is the number of boundary components of F'. In this paper we show that the correct such bound is n'+2, give a partial classification of the pairs (M,T) where the bound n'+2 is reached, and show that if the distance d between the boundary slopes of F and F' is at least 6 then the bound n'+2 cannot be reached; this latter fact allows for the short proof of the classification of the pairs (M,T) with M a hyperbolic 3-manifold and d at least 6 to work without change as outlined in the paper cited above.