Toroidal and Klein bottle boundary slopes

摘要

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. By a result of C. Gordon, if (S, partial derivS), (T, partial derivT) is contained in (M, T_0) are incompressible punctured tori with boundary slopes at distance Δ = Δ(partial derivS, partial derivT). then Δ ≤ 8, and the cases where Δ = 6, 7, 8 are very few and classified. We give a simplified proof of this result (or rather, of its reduction process), using an improved estimate for the maximum possible number of mutually parallel negative edges in the graphs of intersection of S and T. We also extend Gordon's result by allowing either S or T to be an essential Klein bottle.