Stick numbers of Montesinos knots

Abstract

Negami found an upper bound on the stick number [Formula: see text] of a nontrivial knot [Formula: see text] in terms of the minimal crossing number [Formula: see text]: [Formula: see text]. Huh and Oh found an improved upper bound: [Formula: see text]. Huh, No and Oh proved that [Formula: see text] for a [Formula: see text]-bridge knot or link [Formula: see text] with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let [Formula: see text] be a knot or link which admits a reduced Montesinos diagram with [Formula: see text] crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then [Formula: see text]. Furthermore, if [Formula: see text] is alternating, then we can additionally reduce the upper bound by [Formula: see text].