A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots

Spun-knots (respectively, spinning tori) in S-4 made from classical 1-knots compose an important class of 2-knots (respectively, embedded tori) contained in S-4. Virtual 1-knots are generalizations of classical 1-knots. We generalize these constructions to the virtual 1-knot case by using what we call, in this paper, the spinning construction of submanifolds. The construction proceeds as follows: For a virtual 1-knot K, take an embedded circle C contained in (a closed oriented surface F) x (a closed interval [0, 1]), where F is called a representing surface in virtual 1-knot theory. Embed F in S 4 by an embedding map f, and let F stand for f(F). Regard the tubular neighborhood of F in S-4 as the result of rotating F x [0,1] around F. Rotate C together then with F x [0,1]. When C boolean AND (F x {0}) = phi, we obtain an embedded torus Q subset of S-4. We prove the following: The embedding type Q in S-4 depends only on K, and does not depend on f. Furthermore, the submanifolds, Q and "the embedded torus made from K by using Satoh's method", of S-4 are isotopic. Fiberwise equivalence of diagrams refers to fiberwise equivalence of tori in 4-space that lie over the diagrams. We prove that two virtual 1-knot diagrams alpha and beta are fiberwise equivalent if and only if alpha and beta are rotational welded equivalent (see the body of the paper for this definition). We generalize the construction in the virtual 1-knot case written in the first paragraph, and we also succeed to make a consistent construction of one-dimensional-higher submanifolds from any virtual two-dimensional knot. Note that Satoh's method says nothing about the virtual 2-knot case. Rourke's interpretation of Satoh's method is that one puts "fiber-circles" on each point of each virtual 1-knot diagram. If there is no virtual branch point in a virtual 2-knot diagram, our way gives such fiber-circle .s to each point of the virtual 2-knot diagram. Furthermore we prove the following: If a virtual 2-knot diagram a has a virtual branch point, a cannot be covered by such fiber-circles. Hence Rourke's method cannot be generalized to the virtual 2-knot case. Only the spinning construction introduced in this paper works for now.