The prism manifold realization problem

TL;DR

This paper classifies which prism manifolds, a specific type of spherical three-manifolds, can be obtained through positive integral surgeries on knots in the three-sphere, completing the realization problem for all spherical manifolds.

Contribution

It provides a complete list of prism manifolds realizable via positive surgeries on knots in S^3 for q<0, filling the last gap in the spherical manifold realization problem.

Findings

Identifies all prism manifolds P(p, q) realizable by positive surgeries with q<0.

Uses methodology similar to Greene's approach for lens spaces.

Completes the classification for D-type spherical manifolds.

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 $P(p,q)$, for a pair of relatively prime integers $p>1$ and $q$. We determine a complete list of prism manifolds $P(p, q)$ that can be realized by positive integral surgeries on knots in $S^3$ when $q<0$. The methodology undertaken to obtain the classification is similar to that of Greene for lens spaces.