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

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.