Dehn Fillings of Knot Manifolds Containing Essential Twice-Punctured Tori

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