Dehn Fillings of Knot Manifolds Containing Essential Twice-Punctured Tori

We show that if a hyperbolic knot manifold M M contains an essential twice-punctured torus F F with boundary slope β \beta and admits a filling with slope α \alpha producing a Seifert fibred space, then the distance between the slopes α \alpha and β \beta is less than or equal to 5 5 unless M M is the exterior of the figure eight knot. The result is sharp; the bound of 5 5 can be realized on infinitely many hyperbolic knot manifolds. We also determine distance bounds in the case that the fundamental group of the α \alpha -filling contains no non-abelian free group. The proofs are divided into the four cases F F is a semi-fibre, F F is a fibre, F F is non-separating but not a fibre, and F F is separating but not a semi-fibre, and we obtain refined bounds in each case.