Neighbors of Seifert surgeries on a trefoil knot in the Seifert Surgery Network
Arnaud Deruelle, Katura Miyazaki, Kimihiko Motegi

TL;DR
This paper investigates the family of Seifert surgeries derived from a trefoil knot through twisting along seiferters, revealing a rich variety of hyperbolic knots and infinite families of surgeries with specific properties.
Contribution
It characterizes the neighbors of Seifert surgeries on a trefoil knot obtained via twisting along seiferters, highlighting their diversity and the existence of infinitely many special cases.
Findings
Existence of triples of Seifert surgeries on hyperbolic knots for most m values.
Presence of infinitely many Seifert surgeries on strongly invertible hyperbolic knots.
Identification of surgeries not arising from primitive/Seifert-fibered construction.
Abstract
A Seifert surgery is a pair (K, m) of a knot K in the 3-sphere 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 which is not -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.
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Taxonomy
TopicsGeometric and Algebraic Topology · Connective tissue disorders research · Logic, programming, and type systems
