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We show there exists a linear function w: N->N with the following property.Let K be a hyperbolic knot in a hyperbolic 3-manifold M admitting anon-longitudinal S^3 surgery. If K is put into thin position with respect to astrongly irreducible, genus g Heegaard splitting of M then K intersects a thicklevel at most 2w(g) times. Typically, this shows that the bridge number of Kwith respect to this Heegaard splitting is at most w(g), and the tunnel numberof K is at most w(g) + g-1.

Bridge number, Heegaard genus and non-integral Dehn surgery