Stick numbers of 2-bridge knots and links

Abstract

Negami found an upper bound on the stick number s ( K ) s(K) of a nontrivial knot K K in terms of the minimal crossing number c ( K ) c(K) of the knot, which is s ( K ) ≤ 2 c ( K ) s(K) \leq 2 c(K) . Furthermore, McCabe proved that s ( K ) ≤ c ( K ) + 3 s(K) \leq c(K) + 3 for a 2 2 -bridge knot or link, except in the cases of the unlink and the Hopf link. In this paper we construct any 2 2 -bridge knot or link K K of at least six crossings by using only c ( K ) + 2 c(K)+2 straight sticks. This gives a new upper bound on stick numbers of 2 2 -bridge knots and links in terms of crossing numbers.