#### Volume 13, issue 5 (2013)

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Obtaining genus $2$ Heegaard splittings from Dehn surgery

### Kenneth L Baker, Cameron Gordon and John Luecke

Algebraic & Geometric Topology 13 (2013) 2471–2634
##### 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 $2$.

##### Keywords
Dehn surgery, bridge number, Heegaard splitting
Primary: 57M27