Non degenerating Dehn fillings on genus two Heegaard splittings of knots′ complements

It is Thurston’s result that for a hyperbolic knot K in S³, almost all Dehn fillings on its complement result in hyperbolic 3-manifolds except some exceptional cases. So almost all produced 3-manifolds have the same geometry. It is known that its complement in S³, denoted by E(K), admits a Heegaard splitting. Then it is expected that there is a similar result on Heegaard distance for Dehn fillings. In this paper, Dehn fillings on genus two Heegaard splittings are studied. More precisely, we prove that if the distance of a given genus two Heegaard splitting of E(K) is at least 3, then for any two degenerating slopes on ∂E(K), there is a universal bound of their distance in the curve complex of ∂E(K).