On rack colorings for surface-knot diagrams without branch points

Racks do not give us invariants of surface-knots in general. For example, if a surface-knot diagram has branch points (and a rack which we use satisfies some mild condition), then it admits no rack colorings. In this paper, we investigate rack colorings for surface-knot diagrams without branch points and prove that rack colorings are invariants of S2-knots. We also prove that rack colorings for S2-knots can be interpreted in terms of quandles, and discuss a relationship with regular-equivalences of surface-knot diagrams.