The crossing number and maximal bridge length of a knot diagram. With an appendix by Mark Kidwell. (Q1880054)

In 1985, \textit{Brandt-Lickorish} and \textit{Millett} and independently \textit{Ho} discovered a polynomial inviariant \(Q(K)\) in one variable of a link \textit{K. Kidwell} found that \(\max \text{degree } Q(K)\leq c(D) - d(D)\) where \(c(D)\) is the crossing number of a diagram of the knot (or link) \(K\) and \(d(D)\) its maximal bridge length, i.e., the maximal number of consecutive crossing over- or underpasses. In 1986, \textit{H. Morton} asked whether the latter inequality becomes an equality under `certain' minimizing conditions over all diagrams of K. Kidwell mentioned that the Perko knot (denoted by \(10_{161}\)) would be a promising candidate for this. In this note, the author shows that the latter is false by using a result concerning plane curves, the proof of which is given by Kidwell in the appendix of the paper.