Bridge distance, Heegaard genus, and Exceptional Surgeries

We demonstrate a lower bound on the genus of an essential surface or Heegaard surface in a 3-manifold obtained by non-trivial surgery on a link in terms of the bridge distance of a bridge surface for the link. Consequently, knots with high distance bridge surfaces do not admit non-trivial non-hyperbolic surgeries or non-trivial cosmetic surgeries.

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