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Non-Integral Toroidal Surgery On Hyperbolic Knots In S³

By C. McA. Gordon, Y-Q. Wu and X. Zhang


. It is shown that a hyperbolic knot in S 3 admits at most one nonintegral Dehn surgery producing a toroidal manifold. Let K be a knot in the 3-sphere S 3 and M = MK the complement of an open regular neighborhood of K in S 3 . As usual, the set of slopes on the torus #M (i.e. the set of isotopy classes of essential simple loops on #M) is parameterized by {m/n : m,n # Z, n > 0, (m, n) = 1} # {1/0}, so that 1/0 is the meridian slope and 0/1 is the longitude slope. A slope m/n is called non-integral if n # 2. The manifold obtained by Dehn surgery on S 3 along the knot K (equivalently, Dehn filling on M along the torus #M) with slope m/n, is denoted by M(m/n). Now suppose that K # S 3 is a hyperbolic knot, i.e. the interior of M has a complete hyperbolic metric of finite volume. A basic question in Dehn surgery theory is: when can a surgery on K produce a non-hyperbolic 3-manifold? A special case of this question is: when can a surgery on K produce a toroidal 3-manifo..

Year: 2007
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