Remarks relating to the ascending number and geometrical invariants of a knot

Abstract

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an descending diagram. We study relations between the ascending number and geometrical invariants; the crossing number, the genus and the bridge index. The main aim of this paper is to show that there exists a knot [Formula: see text] such that [Formula: see text] and [Formula: see text], and that there exists a knot [Formula: see text] such that [Formula: see text] and [Formula: see text] for any positive integer [Formula: see text]. We also give an upper bound of the ascending number for a [Formula: see text]-bridge knot.