1-parameter family of 1-knot and invariant of surface-knot

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We study many properties of knotted surfaces in Euclidian 4-space; in particular, we prove the triviality of the quandle cocycle invariant of s roll-spun knot, and the existence of a 2-knot for which any 7-coloring requires at lest 6 colors. On the other hand, a knotted torus in 4-space is closely related to a virtual knot. We also study many properties of virtual knots such as two kinds of crossing numbers, n-writhes, and the upper and lower knot groups.