An upper bound of arc index of links

Abstract

In 1996, Cromwell and Nutt [7] found an upper bound on the arc index which is related to the minimal crossing number and conjectured that the upper bound achieves the lowest possible index for alternating links.CONJECTURE. Let L be any prime link. Then α(L) [les ] c(L)+2. Moreover this inequality is strict if and only if L is not alternating.In this paper, we define a new diagram, called a wheel diagram, of a link and use it to prove this conjecture.