Dehn fillings of knot manifolds containing essential once-punctured tori

Abstract

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 $Δ(α,β)\leq 5$. Furthermore we classify the triples $(M; α,β)$ when $\D(α,β)\geq 4$. More precisely, when $\D(α,β)=5$, 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, $\D(α,β)=4$ if and only if $(M; α,β)$ is the triple $\displaystyle (Wh(\frac{-2n\pm1}{n}); -4, 0)$ 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 $\partial M$ 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 $S^3$ are also given.