On the arc index of an adequate link

Abstract

In 1996, Cromwell and Nutt conjectured that α(L) − c (L) = 2 for a link L if and only if L is alternating. In this paper we calculate that α(L) = c (L) for some non-alternating pretzel links L, define a new invariant ρ(L) of adequate links L and show that for each non-negative integer n, there is a prime adequate knot K such that α(K) − c (K) = −2n. We conjecture that α(L) − c (L) = 2ρ(L) for any adequate link L.