Stick numbers of Montesinos knots

Abstract

Negami found an upper bound on the stick number s(K) of a nontrivial knot K in terms of the minimal crossing number c(K): s(K) <= 2c(K). Huh and Oh found an improved upper bound: s(K) <= 3/2(c(K) + 1). Huh, No and Oh proved that s(K) <= c(K) + 2 for a 2-bridge knot or link K 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 K be a knot or link which admits a reduced Montesinos diagram with c(K) crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then s(K) <= c(K) + 3. Furthermore, if K is alternating, then we can additionally reduce the upper bound by 2.