eBook ISBN:  9781470405151 
Product Code:  MEMO/194/909.E 
List Price:  $73.00 
MAA Member Price:  $65.70 
AMS Member Price:  $43.80 
eBook ISBN:  9781470405151 
Product Code:  MEMO/194/909.E 
List Price:  $73.00 
MAA Member Price:  $65.70 
AMS Member Price:  $43.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 194; 2008; 140 ppMSC: Primary 57
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

Chapters

1. Introduction

2. Preliminary lemmas

3. $\hat \Gamma ^+_a$ has no interior vertex

4. Possible components of $\hat \Gamma ^+_a$

5. The case $n_1$, $n_2 > 4$

6. Kleinian graphs

7. If $n_a = 4$, $n_b \geq 4$ and $\hat \Gamma ^+_a$ has a small component then $\Gamma _a$ is kleinian

8. If $n_a = 4$, $n_b \geq 4$ and $\Gamma _b$ is nonpositive then $\hat \Gamma ^+_a$ has no small component

9. If $\Gamma _b$ is nonpositive and $n_a = 4$ then $n_b \leq 4$

10. The case $n_1 = n_2 = 4$ and $\Gamma _1$, $\Gamma _2$ nonpositive

11. The case $n_a = 4$, and $\Gamma _b$ positive

12. The case $n_a = 2$, $n_b \geq 3$, and $\Gamma _b$ positive

13. The case $n_a = 2$, $n_b > 4$, $\Gamma _1$, $\Gamma _2$ nonpositive, and $\max (w_1 + w_2, w_3 + w_4) = 2n_b  2$

14. The case $n_a = 2$, $n_b > 4$, $\Gamma _1$, $\Gamma _2$ nonpositive, and $w_1 = w_2 = n_b$

15. $\Gamma _a$ with $n_a \leq 2$

16. The case $n_a = 2$, $n_b = 3$ or $4$, and $\Gamma _1$, $\Gamma _2$ nonpositive

17. Equidistance classes

18. The case $n_b = 1$ and $n_a = 2$

19. The case $n_1 = n_2 = 2$ and $\Gamma _b$ positive

20. The case $n_1 = n_2 = 2$ and and both $\Gamma _1$, $\Gamma _2$ nonpositive

21. The main theorems

22. The construction of $M_i$ as a double branched cover

23. The manifolds $M_i$ are hyperbolic

24. Toroidal surgery on knots in $S^3$


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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 \(4\).

Chapters

1. Introduction

2. Preliminary lemmas

3. $\hat \Gamma ^+_a$ has no interior vertex

4. Possible components of $\hat \Gamma ^+_a$

5. The case $n_1$, $n_2 > 4$

6. Kleinian graphs

7. If $n_a = 4$, $n_b \geq 4$ and $\hat \Gamma ^+_a$ has a small component then $\Gamma _a$ is kleinian

8. If $n_a = 4$, $n_b \geq 4$ and $\Gamma _b$ is nonpositive then $\hat \Gamma ^+_a$ has no small component

9. If $\Gamma _b$ is nonpositive and $n_a = 4$ then $n_b \leq 4$

10. The case $n_1 = n_2 = 4$ and $\Gamma _1$, $\Gamma _2$ nonpositive

11. The case $n_a = 4$, and $\Gamma _b$ positive

12. The case $n_a = 2$, $n_b \geq 3$, and $\Gamma _b$ positive

13. The case $n_a = 2$, $n_b > 4$, $\Gamma _1$, $\Gamma _2$ nonpositive, and $\max (w_1 + w_2, w_3 + w_4) = 2n_b  2$

14. The case $n_a = 2$, $n_b > 4$, $\Gamma _1$, $\Gamma _2$ nonpositive, and $w_1 = w_2 = n_b$

15. $\Gamma _a$ with $n_a \leq 2$

16. The case $n_a = 2$, $n_b = 3$ or $4$, and $\Gamma _1$, $\Gamma _2$ nonpositive

17. Equidistance classes

18. The case $n_b = 1$ and $n_a = 2$

19. The case $n_1 = n_2 = 2$ and $\Gamma _b$ positive

20. The case $n_1 = n_2 = 2$ and and both $\Gamma _1$, $\Gamma _2$ nonpositive

21. The main theorems

22. The construction of $M_i$ as a double branched cover

23. The manifolds $M_i$ are hyperbolic

24. Toroidal surgery on knots in $S^3$