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 S4 made from classical 1-knots compose an important class of 2-knots (respectively, embedded tori) contained in S4. 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)×(a closed interval [0,1]), where F is called a representing surface in virtual 1-knot theory. Embed F in S4 by an embedding map f, and let F stand for f(F). Regard the tubular neighborhood of F in S4 as the result of rotating F×[0,1] around F. Rotate C together then with F×[0,1]. When C∩(F×{0})=ϕ, we obtain an embedded torus Q⊂S4. We prove the following: The embedding type Q in S4 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 S4 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 α and β are fiberwise equivalent if and only if α and β 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 α has a virtual branch point, α 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.
