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Toroidal Dehn Fillings on Hyperbolic 3-Manifolds
Cameron McA. Gordon, University of Texas at Austin, TX, and Ying-Qing Wu, University of Iowa, Ames, IA
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Memoirs of the American Mathematical Society
2008; 140 pp; softcover
Volume: 194
ISBN-10: 0-8218-4167-X
ISBN-13: 978-0-8218-4167-9
List Price: US$69 Individual Members: US$41.40
Institutional Members: US\$55.20
Order Code: MEMO/194/909

The authors determine all hyperbolic $$3$$-manifolds $$M$$ admitting two toroidal Dehn fillings at distance $$4$$ or $$5$$. They show that if $$M$$ is a hyperbolic $$3$$-manifold with a torus boundary component $$T_0$$, and $$r,s$$ are two slopes on $$T_0$$ with $$\Delta(r,s) = 4$$ or $$5$$ such that $$M(r)$$ and $$M(s)$$ both contain an essential torus, then $$M$$ is either one of $$14$$ specific manifolds $$M_i$$, or obtained from $$M_1, M_2, M_3$$ or $$M_{14}$$ by attaching a solid torus to $$\partial M_i - T_0$$. All the manifolds $$M_i$$ are hyperbolic, and the authors show that only the first three can be embedded into $$S^3$$. As a consequence, this leads to a complete classification of all hyperbolic knots in $$S^3$$ admitting two toroidal surgeries with distance at least $$4$$.

• Introduction
• Preliminary lemmas
• $$\hat \Gamma_a^+$$ has no interior vertex
• Possible components of $$\hat \Gamma_a^+$$
• The case $$n_1, n_2 > 4$$
• Kleinian graphs
• If $$n_a=4$$, $$n_b \geq 4$$ and $$\hat \Gamma_a^+$$ has a small component then $$\Gamma_a$$ is kleinian
• If $$n_a=4$$, $$n_b \geq 4$$ and $$\Gamma_b$$is non-positive then $$\hat \Gamma_a^+$$ has no small component
• If $$\Gamma_b$$ is non-positive and $$n_a=4$$ then $$n_b \leq 4$$
• The case $$n_1 = n_2 = 4$$ and $$\Gamma_1, \Gamma_2$$ non-positive
• The case $$n_a = 4$$, and $$\Gamma_b$$ positive
• The case $$n_a=2$$, $$n_b \geq 3$$, and $$\Gamma_b$$ positive
• The case $$n_a = 2$$, $$n_b > 4$$, $$\Gamma_1, \Gamma_2$$ non-positive, and $$\text{max}(w_1 + w_2,\,\, w_3 + w_4) = 2n_b-2$$
• The case $$n_a = 2$$, $$n_b > 4$$, $$\Gamma_1, \Gamma_2$$ non-positive, and $$w_1 = w_2 = n_b$$
• $$\Gamma_a$$ with $$n_a \leq 2$$
• The case $$n_a = 2$$, $$n_b=3$$ or $$4$$, and $$\Gamma_1,\Gamma_2$$ non-positive
• Equidistance classes
• The case $$n_b = 1$$ and $$n_a = 2$$
• The case $$n_1 = n_2 = 2$$ and $$\Gamma_b$$ positive
• The case $$n_1 = n_2 = 2$$ and both $$\Gamma_1, \Gamma_2$$ non-positive
• The main theorems
• The construction of $$M_i$$ as a double branched cover
• The manifolds $$M_i$$ are hyperbolic
• Toroidal surgery on knots in $$S^3$$
• Bibliography