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