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

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 ≠ - 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.