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

Abstract

Spun-knots (respectively, spinning tori) in [Formula: see text] made from classical 1-knots compose an important class of 2-knots (respectively, embedded tori) contained in [Formula: see text]. 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 [Formula: see text], take an embedded circle [Formula: see text] contained in (a closed oriented surface [Formula: see text])×(a closed interval [Formula: see text]), where [Formula: see text] is called a representing surface in virtual 1-knot theory. Embed [Formula: see text] in [Formula: see text] by an embedding map [Formula: see text], and let [Formula: see text] stand for [Formula: see text] Regard the tubular neighborhood of [Formula: see text] in [Formula: see text] as the result of rotating [Formula: see text] around [Formula: see text]. Rotate [Formula: see text] together then with [Formula: see text]. When [Formula: see text], we obtain an embedded torus [Formula: see text]. We prove the following: The embedding type [Formula: see text] in [Formula: see text] depends only on [Formula: see text], and does not depend on [Formula: see text]. Furthermore, the submanifolds, [Formula: see text] and “the embedded torus made from [Formula: see text] by using Satoh’s method”, of [Formula: see text] 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 [Formula: see text] and [Formula: see text] are fiberwise equivalent if and only if [Formula: see text] and [Formula: see text] 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-circles to each point of the virtual 2-knot diagram. Furthermore we prove the following: If a virtual 2-knot diagram [Formula: see text] has a virtual branch point, [Formula: see text] 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.