On two-generator satellite knots

TL;DR

This paper develops algebraic techniques to analyze the geometric structure of certain 3-manifolds and classifies satellite knots in the 3-sphere with specific two-generator presentations.

Contribution

It introduces methods to determine geometric structures from algebraic data and classifies a class of satellite knots with two-generator presentations.

Findings

Classified satellite knots with specific two-generator presentations in the 3-sphere.

Developed algebraic techniques for analyzing non-simple 3-manifolds.

Connected algebraic data to geometric structures in 3-manifolds.

Abstract

Techniques are introduced which determine the geometric structure of non-simple two-generator $3$-manifolds from purely algebraic data. As an application, the satellite knots in the $3$-sphere with a two-generator presentation in which at least one generator is represented by a meridian for the knot are classified.