On lassos and the Jones polynomial of satellite knots

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.

25 pages, many pictures, comments welcome