Fibered Knots and Virtual Knots
Fibered Knots and Virtual Knots
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纤维结和虚拟结
DOI：
10.1142/s0218216513410034
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
V. Manturov
中科院分区：
文献类型：
--
作者：
M. Chrisman;V. Manturov
介绍了一种用虚纽结理论研究经典纽结的新方法。设$K$是一个纽结，$J$是$K$的补数中的纽结，其中$\text{lk}（J，K）=0$。设有覆盖空间$\pi_J：\Sigma \times（0，1）\to \bar{S^3\backslash V（J）}$，其中$V（J）$是$J$的正则邻域，满足$V（J）\cap \text{im}（K）=\emptyset$，$\Sigma$是连通紧致可定向2-流形。设$K '$是$\Sigma \times（0，1）$中的一个纽结，使得$\pi_J（K'）=K$。然后$K '$稳定到一个虚结$\hat{K}$，称为$K$相对于$J$的虚覆盖。我们调查什么可以说一个经典的结从其虚拟覆盖的情况下，$J$是一个纤维结。几个例子和应用程序的经典节点。建立了虚拟覆盖的基本理论。
We introduce a new technique for studying classical knots with the methods of virtual knot theory. Let $K$ be a knot and $J$ a knot in the complement of $K$ with $\text{lk}(J,K)=0$. Suppose there is covering space $\pi_J: \Sigma \times (0,1) \to \bar{S^3\backslash V(J)}$, where $V(J)$ is a regular neighborhood of $J$ satisfying $V(J) \cap \text{im}(K)=\emptyset$ and $\Sigma$ is a connected compact orientable 2-manifold. Let $K'$ be a knot in $\Sigma \times (0,1)$ such that $\pi_J(K')=K$. Then $K'$ stabilizes to a virtual knot $\hat{K}$, called a virtual cover of $K$ relative to $J$. We investigate what can be said about a classical knot from its virtual covers in the case that $J$ is a fibered knot. Several examples and applications to classical knots are presented. A basic theory of virtual covers is established.