  We develop a construction suggested by Scharlemann and Thompson to obtain an
infinite family of pairs of knots $K_{\alpha}$ and $K'_{\alpha}$ so that
$w(K_{\alpha} # K'_{\alpha})=max{w(K_{\alpha}), w(K'_{\alpha})}$. This is the
first known example of a pair of knots such that $w(K#K')<w(K)+w(K')-2$ and it
establishes that the lower bound $w(K#K')\geq max{w(K),w(K')}$ obtained by
Scharlemann and Schultens is best possible. Furthermore, the knots $K_{\alpha}$
provide an example of knots where the number of critical points for the knot in
thin position is greater than the number of critical points for the knot in
bridge position.
