Berge conjecture

Let $K \subset S^3$ be a knot, let $\gamma$ be a slope on $\partial N(K)$, where $N(K)$ is a tubular neighbourhood of $K$, and suppose that the Dehn surgery $S^3_\gamma(K)$ obtained by filling $S^3 \setminus \operatorname{int} N(K)$ along $\gamma$ is a lens space, i.e. a closed orientable $3$-manifold admitting a genus one Heegaard splitting.

Call $K$ doubly primitive if there is a genus two Heegaard surface $S$ for $S^3$, bounding handlebodies $H_1$ and $H_2$ with $S^3 = H_1 \cup_S H_2$, such that $K \subset S$ and, for $i = 1, 2$, there is a meridian disc $D_i \subset H_i$ (a properly embedded disc whose boundary is essential in $S$) with $K$ meeting $D_i$ transversely in exactly one point. Equivalently, $K$ represents a generator of $\pi_1(H_i)$ for each $i$ after being pushed into $H_i$ off some complete meridian system.

Then $K$ is doubly primitive, and the slope $\gamma$ is the surface slope of $K$ on such a surface $S$, i.e. $\gamma$ is the slope on $\partial N(K)$ determined by the annulus $S \cap \partial N(K)$ pushed off $K$ in $S$.

Here slopes are isotopy classes of unoriented essential simple closed curves on $\partial N(K)$, parametrized as $p/q \in \mathbb{Q} \cup \{1/0\}$ by the standard meridian–longitude basis.