Obtaining genus 2 Heegaard splittings from Dehn surgery (Q357695)

Let \(M=K'(\gamma)\) be the manifold obtained by Dehn surgery on a knot \(K'\) in \(S^3\) along a slope \(\gamma\). Let \(K\) be the core of the attached solid torus. Then \(K\) is considered a knot in \(M\). The authors are interested in the relation between \(K\) and the Heegaard splittings of \(M\). In the case where \(K'\) is hyperbolic, if the Dehn surgery on \(K'\) has distance at least 3 from the meridian and if the 3-manifold \(M\) is of Heegaard genus 2 and does not contain an embedded closed non-orientable 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 where \(M\) does not contain an embedded surface as above, the authors also obtain similar results. As a corollary, they prove that if \(M\) does not contain an incompressible surface of genus 2, then the tunnel number of \(K'\) is at most 2.