Ribbon 2-knots of ribbon crossing number four

Abstract

A [Formula: see text]-knot is a surface in [Formula: see text] that is homeomorphic to [Formula: see text], the standard sphere in [Formula: see text]-space. A ribbon [Formula: see text]-knot is a [Formula: see text]-knot obtained from [Formula: see text] [Formula: see text]-spheres in [Formula: see text] by connecting them with [Formula: see text] pipes. Let [Formula: see text] be a ribbon 2-knot. The ribbon crossing number, denoted by [Formula: see text]-[Formula: see text], is a numerical invariant of the ribbon [Formula: see text]-knot [Formula: see text]. In [T. Yasuda, Crossing and base numbers of ribbon 2-knots, J. Knot Theory Ramifications 10 (2001) 999–1003] we showed that there exist just [Formula: see text] ribbon [Formula: see text]-knots of the ribbon crossing number up to three. In this paper, we show that there exist no more than [Formula: see text] ribbon [Formula: see text]-knots of ribbon crossing number four.