Intersectional pairs of $n$-knots, local moves of $n$-knots, and their associated invariants of $n$-knots

TL;DR

This paper characterizes when pairs of n-knots in spheres, intersecting transversely with PL-homeomorphic intersections, can be realized as such pairs, advancing understanding of their geometric and topological properties.

Contribution

It provides a complete characterization of realizable pairs of n-knots with transverse intersections in spheres, a problem previously unresolved.

Findings

Characterization of realizable pairs of n-knots

Conditions for transverse intersection to be realizable

Advancement in understanding n-knot embeddings

Abstract

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.