Toroidal Dehn Fillings on Hyperbolic 3-manifolds
American Mathematical Society (Verlag)
978-0-8218-4167-9 (ISBN)
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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 $S3$. As a consequence, this leads to a complete classification of all hyperbolic knots in $S3$ 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.
| Erscheint lt. Verlag | 1.8.2008 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Zusatzinfo | illustrations |
| Verlagsort | Providence |
| Sprache | englisch |
| Gewicht | 239 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| ISBN-10 | 0-8218-4167-X / 082184167X |
| ISBN-13 | 978-0-8218-4167-9 / 9780821841679 |
| Zustand | Neuware |
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