The prism manifold realization problem III

TL;DR

This paper completes the classification of prism manifolds realizable by positive integral surgeries on knots in S^3, covering the case when 0<q<p, thus solving the realization problem for all prism manifolds.

Contribution

It provides a complete solution to the prism manifold realization problem by analyzing the case when 0<q<p, building on previous work for other cases.

Findings

Complete classification of prism manifolds realizable by positive integral surgeries.

Identification of all prism manifolds P(p, q) with 0<q<p that can be realized.

Closure of the realization problem for all prism manifolds.

Abstract

Every prism manifold can be parametrized by a pair of relatively prime integers $p>1$ and $q$. In our earlier papers, we determined a complete list of prism manifolds $P(p, q)$ that can be realized by positive integral surgeries on knots in $S^3$ when $q<0$ or $q>p$; in the present work, we solve the case when $0<q<p$. This completes the solution of the realization problem for prism manifolds.