Obtaining genus $2$ Heegaard splittings from Dehn surgery

Abstract

Let $K^{\prime }$ be a hyperbolic knot in $S^3$ and suppose that some Dehn surgery on $K^{\prime }$ with distance at least $3$ from the meridian yields a $3$–manifold $M$ of Heegaard genus $2$. We show that if $M$ does not contain an embedded Dyck’s surface (the closed nonorientable surface of Euler characteristic $-1$), then the knot dual to the surgery is either $0$–bridge or $1$–bridge with respect to a genus $2$ Heegaard splitting of $M$. In the case that $M$ does contain an embedded Dyck’s surface, we obtain similar results. As a corollary, if $M$ does not contain an incompressible genus $2$ surface, then the tunnel number of $K^{\prime }$ is at most $2$.