Width is not additive (Q1945745)

If a knot \(K\) in the \(3\)-sphere is in general position with respect to the standard height function, then the width of \(K\) with respect to the height function is the sum of the number of knot points generically taken at all regular values of the height function. The minimum, taken over all knots isotopic to \(K\), is the width \(w(K)\) of the knot \(K\). As for the behavior of the width with respect to knot sum, it is known that \(w(K \# K') \leq w(K) + w(K') - 2\) (subadditivity) and \(w(K \# K') \geq max\{w(K),w(K')\}\), the latter being a result by \textit{M. Scharlemann} and \textit{J. Schultens} [Trans. Am. Math. Soc. 358, No. 9, 3781--3805 (2006; Zbl 1102.57004)]. The authors find examples showing that the first inequality is strict and that the second inequality is best possible. A further result is that thin position and bridge position for a knot do not coincide.