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For closed, orientable 3-manifolds, Heegaard splittings can be measured by how compressible they are to both sides simultaneously. Hempel (2001) devised the notion of distance to measure this compressibility, and we have numerous results concerning how to bound it. The results to have in mind for this talk are (1) K. Hartshorn (2002) showing that the distances of all Heegaard splittings of Haken 3-manifolds are uniformly bounded and (2) Scharlemann and Tomova (2008) showing that strongly irreducible Heegaard splittings of the same (compact) manifold can be used to bound each others' distances.

For knot complements in the 3-sphere, Heegaard splittings can be produced using circle-valued Morse functions. These "circular Heegaard splittings" have Heegaard surfaces that are disconnected and have boundary so that the handlebodies in the decomposition have distinguished boundary components that present difficulties in the development of a notion of distance. In joint work with P. Weed, we have defined a notion of "circular distance" for these circular Heegaard splittings. We obtain a Hartshorn-type result that provides a uniform bound on the circular distances of all circular Heegaard splittings for a given knot complement in the 3-sphere. In the speaker's dissertation, a result analogous to Scharlemann-Tomova's has been shown and will also be presented.