In 1987, Andrew Casson and Cameron Gordon proved that a certain class of 3-manifolds admit strongly irreducible Heegaard splittings. These 3-manifolds were constructed by beginning with a weakly reducible Heegaard splitting and performing Dehn surgery on a sufficiently complicated knot lying in the splitting surface. John Hempel generalized the notion of strong irreducibility in 2001 by defining the distance of a Heegaard splitting, with strong irreducibility equivalent to distance greater than or equal to 2. Subsequent results have demonstrated a strong connection between this distance and the topology of the ambient 3-manifold.

In the same paper, Hempel proved the existence of Heegaard splittings of arbitrarily high distance by using a high power of a pseudo-Anosov map. Later findings showed that high distance Heegaard splittings could also be obtained via Dehn surgery. These methods start with a Heegaard splitting of some 3-manifold (often the double of a handlebody), and then by an iterative process construct a knot K lying in the splitting surface such that Dehn surgery on K produces a Heegaard splitting of high distance. These results seemed to indicate that all such knots K that admit a high distance splitting after generic Dehn surgery should satisfy some general criterion.

We provide a sufficient condition for Dehn surgery on a knot K to produce a high distance Heegaard splitting. This result can be stated as a natural extension of the work of Casson and Gordon: by picking a knot K of sufficient complexity in the splitting surface, then there is a lower bound on the distance of the Heegaard splitting obtained via Dehn surgery on K that is dependent on the choice of slope for the surgery. We also give an upper bound on the distance of any Heegaard splitting obtained in this manner that depends on the complexity of K. Combining these two bounds, we show that for a fixed K and a generic choice of slope, we can bound the distance of the constructed Heegaard splitting within a range of 2 and, in certain special cases, determine the splitting's exact distance. This is currently the only method of obtaining Heegaard splittings with a known exact high distance that does not require pseudo-Anosov maps.