A Distance for Circular Heegaard Splittings

For a knot $K\subset S^3$, its exterior $E(K)=S^3\mathrm{\backslash }\eta (K)$ has a singular foliation by Seifert surfaces of $K$ derived from a circle-valued Morse function $f\text{\hspace{0.17em}⁣}:E(K)\to S^1$. When $f$ is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critical points decompose $E(K)$ into a pair of compression bodies. We call such a decomposition a circular Heegaard splitting of $E(K)$. We define the notion of circular distance (similar to Hempel distance) for this class of Heegaard splitting and show that it can be bounded under certain circumstances. Specifically, if the circular distance of a circular Heegaard splitting is too large: (1) $E(K)$ can’t contain low-genus incompressible surfaces, and (2) a minimal-genus Seifert surface for $K$ is unique up to isotopy.