# Obtaining genus 2 Heegaard splittings from Dehn surgery

@article{Baker2013ObtainingG2, title={Obtaining genus 2 Heegaard splittings from Dehn surgery}, author={Kenneth L. Baker and Cameron McA. Gordon and John S Luecke}, journal={Algebraic \& Geometric Topology}, year={2013}, volume={13}, pages={2471-2634} }

Let K' be a hyperbolic knot in S^3 and suppose that some Dehn surgery on K' 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 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 M does contain an embedded Dyck's surface, we obtain similar results. As aâ€¦Â

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## References

SHOWING 1-10 OF 46 REFERENCES

Bridge number, Heegaard genus and non-integral Dehn surgery

- Mathematics
- 2012

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 a non-longitudinal S^3 surgery. If K is put into thinâ€¦

Distance of Heegaard splittings of knot complements

- Mathematics
- 2007

Let K be a knot in a closed orientable irreducible 3-manifold M and let P be a Heegaard splitting of the knot complement of genus at least two. Suppose Q is a bridge surface for K. Then eitherâ€¦

The maximal number of exceptional Dehn surgeries

- Mathematics
- 2008

Thurstonâ€™s hyperbolic Dehn surgery theorem is one of the most important results in 3-manifold theory, and it has stimulated an enormous amount of research. If M is a compact orientable hyperbolicâ€¦

A Seifert Fibered Manifold with Infinitely Many Knot-Surgery Descriptions

- Mathematics
- 2010

Osoinach introduced a way to construct a 3-manifold which can be obtained by the same integral Dehn surgery on an infinite number of knots in the 3-sphere. Using it, he gave such a hyperbolicâ€¦

On nonsimple 3-manifolds and 2-handle addition

- Mathematics
- 1994

Abstract Suppose a 2-handle is attached to an orientable, irreducible 3-manifold M along a curve Î± contained in âˆ‚ M , obtaining a 3-manifold M Î± . Suppose âˆ‚ M is compressible, but âˆ‚ M â€” Î± is not. Itâ€¦

Dehn surgeries on knots creating essential tori, II

- Mathematics
- 2000

In this paper, which is a sequel to [GLul], we continue our study of when Dehn surgery on a hyperbolic knot K in S can yield a manifold that contains an incompressible torus. Let E(K) denote theâ€¦

Structures of the Haken manifolds with Heegaard splittings of genus two

- Mathematics
- 1984

ÎŒ(n) (A(n), MÏ‹(n) resp.): the collection of the Seifert fibered manifolds the orbit manifold of each of which is a disk (annulus, Mobius band resp.) with n exceptional fibers. MÎº (ML resp.) : theâ€¦

Combinatorial methods in Dehn surgery

- Mathematics
- 1997

This is an expository paper, in which we give a summary of some of the joint work of John Luecke and the author on Dehn surgery. We consider the situation where we have two Dehn fillings M(Î±) andâ€¦

On Culler-Shalen seminorms and Dehn filling

- Mathematics
- 1998

If F is a finitely generated discrete group and G a complex algebraic Lie group, the G-character variety of r is an affine algebraic variety whose points correspond to characters of representationsâ€¦

Persistence of Heegaard structures under Dehn filling

- Mathematics
- 2001

It is well known that a Heegaard surface may destabilize after Dehn filling, reducing the genus by one or more. This phenomenon is classified according to whether or not the core of the attachedâ€¦