An upper bound of arc index of links

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