The prism manifold realization problem

Abstract

The spherical manifold realization problem asks which spherical three-manifolds arise from surgeries on knots in $S^3$. In recent years, the realization problem for C–, T–, O– and I–type spherical manifolds has been solved, leaving the D–type manifolds (also known as the prism manifolds) as the only remaining case. Every prism manifold can be parametrized as $P\left(p,q\right)$ for a pair of relatively prime integers $p>1$ and $q$. We determine a list of prism manifolds $P\left(p,q\right)$ that can possibly be realized by positive integral surgeries on knots in $S^3$ when $q<0$. Based on the forthcoming work of Berge and Kang, we are confident that this list is complete. The methodology undertaken to obtain the classification is similar to that of Greene for lens spaces.