| eBook ISBN: | 978-1-4704-0515-1 |
| Product Code: | MEMO/194/909.E |
| List Price: | $73.00 |
| MAA Member Price: | $65.70 |
| AMS Member Price: | $43.80 |
| eBook ISBN: | 978-1-4704-0515-1 |
| Product Code: | MEMO/194/909.E |
| List Price: | $73.00 |
| MAA Member Price: | $65.70 |
| AMS Member Price: | $43.80 |
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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 \(4\).
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Table of Contents
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Chapters
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1. Introduction
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2. Preliminary lemmas
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3. $\hat \Gamma ^+_a$ has no interior vertex
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4. Possible components of $\hat \Gamma ^+_a$
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5. The case $n_1$, $n_2 > 4$
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6. Kleinian graphs
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7. If $n_a = 4$, $n_b \geq 4$ and $\hat \Gamma ^+_a$ has a small component then $\Gamma _a$ is kleinian
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8. If $n_a = 4$, $n_b \geq 4$ and $\Gamma _b$ is non-positive then $\hat \Gamma ^+_a$ has no small component
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9. If $\Gamma _b$ is non-positive and $n_a = 4$ then $n_b \leq 4$
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10. The case $n_1 = n_2 = 4$ and $\Gamma _1$, $\Gamma _2$ non-positive
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11. The case $n_a = 4$, and $\Gamma _b$ positive
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12. The case $n_a = 2$, $n_b \geq 3$, and $\Gamma _b$ positive
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13. The case $n_a = 2$, $n_b > 4$, $\Gamma _1$, $\Gamma _2$ non-positive, and $\max (w_1 + w_2, w_3 + w_4) = 2n_b - 2$
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14. The case $n_a = 2$, $n_b > 4$, $\Gamma _1$, $\Gamma _2$ non-positive, and $w_1 = w_2 = n_b$
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15. $\Gamma _a$ with $n_a \leq 2$
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16. The case $n_a = 2$, $n_b = 3$ or $4$, and $\Gamma _1$, $\Gamma _2$ non-positive
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17. Equidistance classes
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18. The case $n_b = 1$ and $n_a = 2$
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19. The case $n_1 = n_2 = 2$ and $\Gamma _b$ positive
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20. The case $n_1 = n_2 = 2$ and and both $\Gamma _1$, $\Gamma _2$ non-positive
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21. The main theorems
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22. The construction of $M_i$ as a double branched cover
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23. The manifolds $M_i$ are hyperbolic
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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 non-positive then $\hat \Gamma ^+_a$ has no small component
-
9. If $\Gamma _b$ is non-positive and $n_a = 4$ then $n_b \leq 4$
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10. The case $n_1 = n_2 = 4$ and $\Gamma _1$, $\Gamma _2$ non-positive
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11. The case $n_a = 4$, and $\Gamma _b$ positive
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12. The case $n_a = 2$, $n_b \geq 3$, and $\Gamma _b$ positive
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13. The case $n_a = 2$, $n_b > 4$, $\Gamma _1$, $\Gamma _2$ non-positive, and $\max (w_1 + w_2, w_3 + w_4) = 2n_b - 2$
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14. The case $n_a = 2$, $n_b > 4$, $\Gamma _1$, $\Gamma _2$ non-positive, and $w_1 = w_2 = n_b$
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15. $\Gamma _a$ with $n_a \leq 2$
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16. The case $n_a = 2$, $n_b = 3$ or $4$, and $\Gamma _1$, $\Gamma _2$ non-positive
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17. Equidistance classes
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18. The case $n_b = 1$ and $n_a = 2$
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19. The case $n_1 = n_2 = 2$ and $\Gamma _b$ positive
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20. The case $n_1 = n_2 = 2$ and and both $\Gamma _1$, $\Gamma _2$ non-positive
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21. The main theorems
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22. The construction of $M_i$ as a double branched cover
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23. The manifolds $M_i$ are hyperbolic
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24. Toroidal surgery on knots in $S^3$
