On lassos and the Jones polynomial of satellite knots

TL;DR

This thesis introduces lassos in the solid torus, constructs satellite knots with specific Alexander polynomials, and provides a formula for their Jones polynomials, revealing distinctions for certain parameters.

Contribution

It defines a new family of knots called lassos, and derives a formula for the Jones polynomial of satellite knots using these lassos as patterns.

Findings

Satellite knots with certain parameters have distinct Jones polynomials.

Lassos enable controlled construction of satellite knots with desired Alexander polynomials.

A new formula relates the Jones polynomial of satellite knots to their pattern and companion.

Abstract

In this master thesis, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot $K$ with Alexander polynomial $\Delta_K(t)$, I then use these lassos as patterns to construct families of satellite knots that have Alexander polynomial $\Delta_K(t^d)$ where $d\in\mathbb{N}\cup \{0\}$. In particular, I prove that if $d\in\{0,1,2,3\}$ these satellite knots have different Jones polynomials. For this purpose, I give rise to a formula for calculating the Jones polynomial of a satellite knot in terms of the Jones polynomials of its pattern and companion.