Ribbon crossing numbers, crossing numbers, and Alexander polynomials

A 2-knot is a surface in R4 that is homeomorphic to S2, the standard sphere in 3-space. A ribbon 2-knot is a 2-knot obtained from m 2-spheres in R4 by connecting them with m−1 annuli. Let K2 be a ribbon 2-knot. The ribbon crossing number, denoted by r-cr(K2) is a numerical invariant of the ribbon 2-knot K2. It is known that the degree of the Alexander polynomial of K2 is less than or equal to r-cr(K2). In this paper, we show that r-cr(K2) is estimated by coefficients in the Alexander polynomial of K2. Furthermore, applying this fact, for a classical knot k1, we also estimate the crossing number, denoted by cr(k1).