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