Networking Seifert Surgeries on Knots IV: Seiferters and branched coverings

TL;DR

This paper explores the structure of Seifert surgeries on knots, introducing methods to find seiferters and paths in the Seifert Surgery Network, especially for surgeries derived from branched covers, connecting complex surgeries to simpler torus knot surgeries.

Contribution

It provides a new method to identify seiferters for Seifert surgeries obtained via branched covers and constructs explicit paths in the Seifert Surgery Network to torus knot surgeries.

Findings

Explicit paths from certain Seifert surgeries to torus knot surgeries.

Method to find seiferters for surgeries from branched covers.

Identification of infinite families of surgeries connected in the network.

Abstract

A Seifert surgery is an integral surgery on a knot in S^3 producing a Seifert fiber space which may contain an exceptional fiber of index 0. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; its edges correspond to single twistings along "seiferters" or "annular pairs of seiferters". One problem of the network is whether there is a path from each vertex to a vertex on a torus knot, the most basic Seifert surgery. We give a method to find seiferters and annular pairs of seiferters for Seifert surgeries obtained by taking two--fold branched covers of tangles. Concerning three infinite families of Seifert surgeries obtained by the second author via branched covers, we find explicit paths in the network from such surgeries to Seifert surgeries on torus knots.