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

• 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