虚拟1-节和2-节的纺纱结构以及虚拟1-节的纤维和焊接等效性
A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots
摘要：
我们成功地推广自旋结的经典1-节的虚拟1-节的情况下，通过使用“自旋构造”。也就是说，我们证明了以下几点：让$Q$是一个虚拟的1-knot $K$通过我们的方法纺结。$S^4$中的嵌入类型$Q$仅取决于$K$。进一步证明了：$S^4$中的子流形$Q$和由Satoh方法定义的$K,$构成的嵌入环面是同位素的。我们成功地将上述构造推广到了虚2节情形。请注意，Satoh的方法没有提到虚拟2节的情况。鲁尔克对佐藤方法的解释是，人们在每个虚拟1-节图的每个点上放置“纤维圈”。如果虚2-纽结图中没有虚分支点，我们的方法将这样的纤维圆赋予虚2-纽结图中的每个点。我们证明了：如果一个虚拟2-节点图$α$有一个虚拟分支点，$α$不能覆盖这样的纤维圈。利用我们的旋转构造，我们得到了虚2-纽结图集合的一个新的等价关系，即$\mathcal E$ -等价关系。我们证明，有虚拟2结图，几乎是不等价的，但$\mathcal E$ -等价。虽然Rourke声称两个虚1-节图$α$和$β$是纤维等效的当且仅当$α$和$β$是焊接等效的，我们声明这种说法是错误的。我们证明了两个虚1-节图$α$和$β$是纤维方向等价的当且仅当$α$和$β$是旋转焊接等价的。
英文摘要：
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 $α$ has a virtual branch point, $α$ 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 $α$ and $β$ are fiberwise equivalent if and only if $α$ and $β$ are welded equivalent, we state that this claim is wrong. 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.
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