Companions of the unknot and width additivity

Abstract

It has been conjectured that for knots $K$ and $K^{\prime }$ 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 $K_i^n$ 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 $K_i^n$ produces such counterexamples.

12 pages, 11 figures