The prism manifold realization problem
The spherical manifold realization problem asks which spherical three-manifolds arise from surgeries on knots in S³. 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 list of prism manifolds P(p,q) that can possibly be realized by positive integral surgeries on knots in S³ 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.
© 2020 Mathematical Sciences Publishers. Received: 17 May 2018; Revised: 6 June 2019; Accepted: 24 June 2019; Published: 23 April 2020. Ni was partially supported by NSF grant number DMS-1252992 and an Alfred P Sloan Research Fellowship. Ballinger, Hsu, Mackey and Ochse were supported by Caltech's Summer Undergraduate Research Fellowships (SURF) program. Ballinger also wishes to thank Samuel P and Frances Krown for their generous support through the SURF program. We are grateful to John Berge for sending us the preprint  and some useful programs. We thank Zhengyuan Shang for finding a typo in Table 3. We thank the referee for a very thorough review.
Submitted - 1612.04921.pdf
Published - agt-v20-n2-p06-s.pdf