Width of a satellite knot and its companion

TL;DR

This paper proves a conjecture relating the width of satellite knots to their companions, establishing a quadratic lower bound and identifying cases of equality for braid pattern satellites.

Contribution

It proves Zupan's conjecture on the width of satellite knots and characterizes when equality holds for braid pattern satellites.

Findings

Proved that the width of a satellite knot is at least n^2 times the width of its companion.

Established that equality holds for satellite knots with braid pattern.

Confirmed the conjecture by Alexander Zupan.

Abstract

In the paper we prove the conjecture by Alexander Zupan that $w(K) \geqslant n^2w(J)$ where w denote the width and $K$ and $J$ are satellite knot and its companion with winding number $n$. Also we proved that for satellite knot with braid pattern, the equality holds.