Lens space surgeries & primitive/Seifert type constructions

TL;DR

This paper demonstrates that certain lens space surgeries on knots in the 3-sphere, originating from primitive/Seifert constructions, also originate from primitive/primitive constructions, advancing the understanding of the Berge conjecture.

Contribution

It establishes a link between primitive/Seifert and primitive/primitive constructions for lens space surgeries, aiding the proof of the Berge conjecture for tunnel number one knots.

Findings

Lens space surgeries from primitive/Seifert also come from primitive/primitive constructions.

Progress towards proving the Berge conjecture for tunnel number one knots.

First step in a three-step program for the Berge conjecture.

Abstract

We show that lens space surgeries on knots in $S^3$ which arise from the primitive/Seifert type construction also arise from the primitive/primitive construction. This is the first step of a three step program to prove the Berge conjecture for tunnel number one knots.