Companions of the unknot and width additivity

TL;DR

This paper investigates the additivity of knot width under connected sum, confirming the conjecture for certain families of knots and ruling out specific counterexamples for knots with bridge number greater than two.

Contribution

The paper proves that for knots with bridge number greater than two, the proposed counterexamples to width additivity do not occur.

Findings

Confirmed the conjecture for knots with bridge number > 2.

Ruled out specific counterexamples proposed by Scharlemann and Thompson.

Supported the additivity of knot width in the studied cases.

Abstract

It has been conjectured that for knots $K$ and $K'$ in $S^3$, $w(K#K')= w(K)+w(K')-2$. Scharlemann and Thompson have proposed potential counterexamples to this conjecture. For every $n$, they proposed a family of knots ${K^n_i}$ for which they conjectured that $w(B^n#K^n_i)=w(K^n_i)$ where $B^n$ is a bridge number $n$ knot. We show that for $n>2$ none of the knots in ${K^n_i}$ produces such counterexamples.