  For a knot $K\subset S^3$, its exterior $E(K) = S^3\backslash\eta(K)$ has a
singular foliation by Seifert surfaces of $K$ derived from a circle-valued
Morse function $f\colon E(K)\to S^1$. 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.
