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

TL;DR

This paper extends spinning constructions to virtual 1- and 2-knots, establishing new equivalence relations and clarifying the relationship between fiberwise and welded equivalences, with implications for virtual knot theory.

Contribution

The authors generalize spun knots to virtual cases, introduce the $\\\ extbf{\E}$-equivalence, and clarify the relationship between fiberwise and welded equivalences for virtual 1-knots.

Findings

Spun knots of virtual 1-knots depend only on the original knot.

Virtual 2-knots with virtual branch points cannot be fibered with fiber-circles.

Fiberwise equivalence for virtual 1-knots is characterized by rotational welded equivalence.

Abstract

We succeed to generalize spun knots of classical 1-knots to the virtual 1-knot case by using the `spinning construction'. That, is, we prove the following: Let $Q$ be a spun knot of a virtual 1-knot $K$ by our method. The embedding type $Q$ in $S^4$ depends only on $K$. Furthermore we prove the following: The submanifolds, $Q$ and the embedded torus made from $K,$ defined by Satoh's method, in $S^4$ are isotopic. We succeed to generalize the above construction to the virtual 2-knot case. 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. We prove the following: If a virtual 2-knot diagram $\alpha$ has a virtual branch point, $\alpha$ cannot be covered by such fiber-circles. We obtain a new equivalence relation, the $\mathcal E$-equivalence relation of the set of virtual 2-knot diagrams, by using our spinning construction. We prove that there are virtual 2-knot diagrams that are virtually nonequivalent but are $\mathcal E$-equivalent. Although Rourke claimed that two virtual 1-knot diagrams $\alpha$ and $\beta$ are fiberwise equivalent if and only if $\alpha$ and $\beta$ are welded equivalent, we state that this claim is wrong. 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.