A new bound on odd multicrossing numbers of knots and links

Abstract

An [Formula: see text]-crossing projection of a link [Formula: see text] is a projection of [Formula: see text] onto a plane such that [Formula: see text] points on [Formula: see text] are superimposed on top of each other at every crossing. We prove that for all [Formula: see text] and all links [Formula: see text], the inequality [Formula: see text] holds, where [Formula: see text], [Formula: see text] and [Formula: see text] are the [Formula: see text]-crossing number, [Formula: see text]-genus, and the number of components of [Formula: see text] respectively. This result is used to prove a new bound on the odd crossing numbers of torus knots and generalizes a result of Jablonowski (see [M. Jabłonowski, Triple-crossing number, the genus of a knot or link and torus knots, Topology Appl. 285 (2020) 107389]). We also prove a new upper bound on the [Formula: see text]-crossing numbers of the 2-torus knots and links. Furthermore, we improve the lower bounds on the [Formula: see text]-crossing numbers of [Formula: see text] knots with [Formula: see text]-crossing number at most [Formula: see text]. Finally, we improve the lower bounds on the [Formula: see text]-crossing numbers of [Formula: see text] knots with [Formula: see text]-crossing number at most [Formula: see text].