Neighbors of Seifert surgeries on a trefoil knot in the Seifert Surgery Network

Abstract

A Seifert surgery is a pair $$(K, m)$$ of a knot $$K$$ in $$S^3$$ and an integer $$m$$ such that $$m$$ -Dehn surgery on $$K$$ results in a Seifert fiber space allowed to contain fibers of index zero. Twisting $$K$$ along a trivial knot called a seiferter for $$(K, m)$$ yields Seifert surgeries. We study Seifert surgeries obtained from those on a trefoil knot by twisting along their seiferters. Although Seifert surgeries on a trefoil knot are the most basic ones, this family is rich in variety. For any $$m \ne -2$$ it contains a successive triple of Seifert surgeries $$(K, m)$$ , $$(K, m +1)$$ , $$(K, m +2)$$ on a hyperbolic knot $$K$$ , e.g. $$17$$ -, $$18$$ -, $$19$$ -surgeries on the $$(-2, 3, 7)$$ pretzel knot. It contains infinitely many Seifert surgeries on strongly invertible hyperbolic knots none of which arises from the primitive/Seifert-fibered construction, e.g. $$(-1)$$ -surgery on the $$(3, -3, -3)$$ pretzel knot.