Intersectional pairs of n-knots, local moves of n-knots, and their associated invariants of n-knots
classification
🧮 math.GT
keywords
knotsspheren-knotspairrealizableassociatedcharacterizationcomplete
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Let $n$ be an integer$\geqq0$. Let $S^{n+2}_1$ (respectively, $S^{n+2}_2$) be the $(n+2)$-sphere embedded in the $(n+4)$-sphere $S^{n+4}$. Let $S^{n+2}_1$ and $S^{n+2}_2$ intersect transversely. Suppose that the smooth submanifold, $S^{n+2}_1 \cap S^{n+2}_2$ in $S^{n+2}_i$ is PL homeomophic to the $n$-sphere. Then $S^{n+2}_1$ and $S^{n+2}_2$ in $S^{n+2}_i$ is an $n$-knot $K_i$. We say that the pair $(K_1,K_2)$ of n-knots is realizable. We consider the following problem in this paper. Let $A_1$ and $A_2$ be n-knots. Is the pair $(A_1,A_2)$ of $n$-knots realizable? We give a complete characterization.
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