A geometric invariant of virtual n-links

Some generalizations of Satoh's Tube map

“…For example, consider creating a virtual m-link by horizontal 1-spinning the virtual trefoil m times. This is a nontrivial virtual m-link, as may be seen from its stack invariants, [14]. However, the welded isotopy which makes the virtual trefoil w-equivalent to the unknot also spins to give a welded isotopy of the m-spun virtual trefoil, turned into a welded m-knot via the T mn map, to an unknotted m-knot.…”

“…In fact, the homotopy type of D(K) is invariant under welded equivalence of virtual 1-links as well. See [14]. We can define the link group G(K) to be the fundamental group of D(K).…”

“…For convenience, we may wish to reverse the orientation of this copy. The vertical double was introduced in [14]. Now interpret V D(K) as a welded m-link with extra structure: observe that the disk fibers (of the CW complex we use in the T mn construction) of V D(K) are naturally paired, with every fiber of K * corresponding to V D(K)(m, 1) contained in the fiber of K coming from V D(K)(m, 0).…”

“…In addition to virtual links, Satoh also defined virtual arc diagrams and defined the T ube map on such diagrams. On the other hand, in [15,14], the notion of virtual knots was extended to arbitrary dimensions, building on the geometric description of virtual knots given by Kuperberg, [7]. It is therefore natural to ask whether Satoh's idea for the T ube map can be generalized into higher dimensions, as well as inquiring into a generalization of the virtual arc diagrams to higher dimensions.…”