Width is not additive

Abstract

We develop the construction suggested by Scharlemann and Thompson in [Proc. of the Casson Fest. (2004) 135-144] to obtain an infinite family of pairs of knots $K_{\alpha }$ and $K_{\alpha }^{\prime }$ so that $w\left(K_{\alpha }\#K_{\alpha }^{\prime }\right)=max\left\{w\left(K_{\alpha }\right),w\left(K_{\alpha }^{\prime }\right)\right\}$. This is the first known example of a pair of knots such that $w\left(K\#K^{\prime }\right)<w\left(K\right)+w\left(K^{\prime }\right)-2$ and it establishes that the lower bound $w\left(K\#K^{\prime }\right)\ge max\left\{w\left(K\right),w\left(K^{\prime }\right)\right\}$ obtained in Scharlemann and Schultens [Trans. Amer. Math. Soc. 358 (2006) 3781-3805] 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.