# Toroidal Dehn Fillings on Hyperbolic 3-Manifolds

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*Cameron McA. Gordon; Ying-Qing Wu*

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 \(4\).

#### Table of Contents

# Table of Contents

## Toroidal Dehn Fillings on Hyperbolic 3-Manifolds

- Contents v6 free
- 1. Introduction 18 free
- 2. Preliminary lemmas 512 free
- 3. Γ[sup(+)][sub(a)] has no interior vertex 1825
- 4. Possible components of Γ[sup(+)][sub(a)] 2027
- 5. The case n1, n2 > 4 2633
- 6. Kleinian graphs 3441
- 7. If n[sub(a)] = 4, n[sub(b)] ≥ 4 and Γ[sup(+)][sub(a)] has a small component then Γ[sub(a)] is kleinian 3744
- 8. If n[sub(a)] = 4, n[sub(b)] ≥ 4 and Γ[sub(b)] is non-positive then Γ[sup(+)][sub(a)] has no small component 4148
- 9. If Γ[sub(b)] is non-positive and n[sub(a)] = 4 then n[sub(b)] ≤ 4 4653
- 10. The case n[sub(1)] = n[sub(2)] = 4 and Γ[sub(1)], Γ[sub(2)] non-positive 5158
- 11. The case n[sub(a)] = 4, and Γ[sub(b)] positive 5461
- 12. The case n[sub(a)] = 2, n[sub(b)] ≥ 3, and Γ[sub(b)] positive 6471
- 13. The case n[sub(a)] = 2, n[sub(b)] > 4, Γ[sub(1)], Γ[sub(2)] non-positive, and max(w[sub(1)] + w[sub(2)], w[sub(3)] + w[sub(4)]) = 2[sub(nb) … 2 7481
- 14. The case n[sub(a)] = 2, n[sub(b)] > 4, Γ[sub(1)], Γ[sub(2)] non-positive, and w[sub(1)] = W[sub(2)] = n[sub(b)] 7885
- 15. Γ[sub(a)] with n[sub(a)] ≤ 2 8592
- 16. The case n[sub(a)] = 2, n[sub(b)] = 3 or 4, and Γ[sub(1)], Γ[sub(2)] non-positive 8693
- 17. Equidistance classes 94101
- 18. The case n[sub(b)] = 1 and n[sub(a)] = 2 96103
- 19. The case n[sub(1)] = n[sub(2)] = 2 and Γ[sub(b)] positive 97104
- 20. The case n[sub(1)] = n[sub(2)] = 2 and and both Γ[sub(1)], Γ[sub(2)] non-positive 103110
- 21. The main theorems 108115
- 22. The construction of M[sub(i)] as a double branched cover 111118
- 23. The manifolds M[sub(i)] are hyperbolic 122129
- 24. Toroidal surgery on knots in S[sup(3)] 131138
- Bibliography 139146