We determine all hyperbolic 3-manifolds М admitting two toroidal Dehn fill-ings at distance 4 or 5. We show that if М is a hyperbolic 3-manifold with a torusboundary component Т_o,and sare two slopes onТ_owith Δ(r, s) = 4 or; 5 suchthat М(r) and М(s) both contain an essential torus, then М is either one of 14specific manifolds М_i,or obtained fromМ_1,М_2, 3orMuby attaching' a solidtorus to М_i —Т_o.All the manifoldsМ_iare hyperbolic, and we show that only thefirst three can be embedded into S~3. As a consequence, this leads to completeclassification of all hyperbolic knots in S~3 admitting two toroidal surgeries withdistance at least 4.
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