Upper bound for the alternation number of a torus knot
\textit{A. Kawauchi} [Topology Appl. 157, No.~1, 274--279 (2010; Zbl 1188.57002)] introduced a numerical invariant for a knot or link, called the alternation number, which is the minimal number of crossing changes needed to deform the knot or link into an alternating one. In the present paper, the author gives estimations for the alternation numbers of torus knots of \(3\)-, \(4\)- and \(5\)-braids, and of some special types. In particular, the alternation numbers are determined for torus knots \(T(3, l)\), \(l \equiv 1, 2\pmod 6\) and \(T(4, 5)\), for which the author uses the inequality relating the alternation number, signature and Rasmussen \(s\)-invariant of a knot, developed by \textit{T. Abe} [J. Knot Theory Ramifications 18, No. 3, 363--379 (2009; Zbl 1180.57005)]. It is also shown that, for any positive integer \(k\), there exist infinitely many \(3\)-braid knots whose alternation number is \(k\).
scientific article; zbMATH DE number 5641774
scientific article