Mp-small summands increase knot width

Abstract

Scharlemann and Schultens have shown that for any pair of knots $K_1$ and $K_2$, $w\left(K_1\#K_2\right)\ge max\left\{w\left(K_1\right),w\left(K_2\right)\right\}$. Scharlemann and Thompson have given a scheme for possible examples where equality holds. Using results of Scharlemann–Schultens, Rieck–Sedgwick and Thompson, it is shown that for $K={\#}_{i=1}^n{K}_i$ a connected sum of mp-small knots and $K^{\prime }$ any non-trivial knot, $w\left(K\#K^{\prime }\right)>w\left(K\right)$.