The prism manifold realization problem

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.Remark 1.3. If we allow r = −1 in Theorem 1.2, we get p = 2q + 1: see Theorem 1.1.

The prism manifold realization problem III

“…Doig, in [7, Conjecture 12], conjectured that if $$P(p,q)$$ is realizable, then $$p\leqslant 2|q|+1$$. The main result of [1] settles the conjecture for $$q&lt;0$$; Theorem 1.1 verifies it for $$q&gt;0$$.…”

“…Let $$P(p,q)$$ be an oriented prism manifold with Seifert invariants $$\begin{equation*} (-1; (2,1), (2,1), (p,q)), \end{equation*}$$where $$q$$ and $$p&gt;1$$ are relatively prime integers. See [1, Section 2] for the convention of Seifert invariants and basic topological properties of prism manifolds. In [1, 2], we solved the Dehn surgery realization problem of prism manifolds for $$q&lt;0$$ and for $$q&gt;p$$.…”

“…See [1, Section 2] for the convention of Seifert invariants and basic topological properties of prism manifolds. In [1, 2], we solved the Dehn surgery realization problem of prism manifolds for $$q&lt;0$$ and for $$q&gt;p$$. The theme of the present work is to settle the remaining case $$0&lt;q&lt;p$$.…”

“…The theme of the present work is to settle the remaining case $$0&lt;q&lt;p$$. In [1, Tables 1 and 2], the authors give a tabulation of prism manifolds that can be obtained by positive integral Dehn surgery on Berge–Kang knots [4]. The tables conjecturally account for all realizable prism manifolds; in particular, [1, Table 2] suggests that for a realizable $$P(p,q)$$ with $$q&gt;0$$, we must have $$p\leqslant 2q+1$$.…”

“…In [1, Tables 1 and 2], the authors give a tabulation of prism manifolds that can be obtained by positive integral Dehn surgery on Berge–Kang knots [4]. The tables conjecturally account for all realizable prism manifolds; in particular, [1, Table 2] suggests that for a realizable $$P(p,q)$$ with $$q&gt;0$$, we must have $$p\leqslant 2q+1$$. Indeed, this is the case: Theorem If $$P(p,q)$$ with $$q&gt;0$$ can be obtained by surgery on a knot $$K\subset S^3$$, then $$p\leqslant 2q+1$$.…”