Dehn Fillings of Knot Manifolds Containing Essential Once-punctured Tori

In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let M be such a knot manifold and let β be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling M with slope α produces a Seifert fibred manifold, then ∆(α, β) ≤ 5. Furthermore we classify the triples (M ;α, β) when ∆(α, β) ≥ 4. More precisely, when ∆(α, β) = 5, then M is the (unique) manifold Wh(−3/2) obtained by Dehn filling one boundary component of the Whitehead link exterior with slope −3/2, and (α, β) is the pair of slopes (−5, 0). Further, ∆(α, β) = 4 if and only if (M ;α, β) is the triple (Wh( −2n± 1 n );−4, 0) for some integer n with |n| > 1. Combining this with known results, we classify all hyperbolic knot manifolds M and pairs of slopes (β, γ) on ∂M where β is the boundary slope of an essential once-punctured torus in M and γ is an exceptional filling slope of distance 4 or more from β. Refined results in the special case of hyperbolic genus one knot exteriors in S are also given.