Toroidal and Klein bottle boundary slopes
Let M be a compact, connected, orientable, irreducible 3-manifold and T0 an incompressible torus boundary component of M such that the pair (M,T0) is not cabled. By a result of C. Gordon, if (S,∂S),(T,∂T ) ⊂ (M,T0) are incompressible punctured tori with boundary slopes at distance Δ = Δ(∂S,∂T ), 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.