# A Distance for Circular Heegaard Splittings

## TL;DR

This paper introduces a new notion of distance for circular Heegaard splittings of knot exteriors, providing bounds that relate to the topology of the knot complement and the uniqueness of Seifert surfaces.

## Contribution

It defines the circular distance for circular Heegaard splittings and establishes bounds on the topology of knot exteriors based on this distance.

## Key findings

- Large circular distance implies no low-genus incompressible surfaces.
- Large circular distance ensures the uniqueness of minimal-genus Seifert surfaces.
- Bounds on the circular distance relate to the topological complexity of knot exteriors.

## Abstract

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.