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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.

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