Scharlemann and Schultens have shown that for any pair of knots and , . 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 a connected sum of mp-small knots and any non-trivial knot, .
"Mp-small summands increase knot width." Algebr. Geom. Topol. 4 (2) 1041 - 1044, 2004. https://doi.org/10.2140/agt.2004.4.1041