Singularities of the projections of $n$-dimensional knots

TL;DR

This paper constructs specific high-dimensional knotted spheres with controlled projection singularities, demonstrating complex behaviors in their projections and establishing new examples of knotted spheres with particular projection properties.

Contribution

It introduces new examples of knotted n-spheres in (n+2)-space with prescribed singularity sets in their projections, advancing understanding of high-dimensional knot projections.

Findings

Existence of knotted n-spheres with exactly two double point components in projection.

Construction of embeddings with projections having no double points or connected double point sets.

Demonstration that certain projections cannot be realized by unknotted spheres.

Abstract

Let n be aninteger>4. There is a smoothly knotted n-dimensional sphere in (n+2)-space such that the singular point set of its projection in (n+1)-space consists of double points and that the components of the singular point set are two. (The sphere is knotted in the sense that it does not bound any embedded (n+1)-ball in (n+2)-space.) Furthermore, the projection is not the projection of any unknotted sphere in the (n+2)-space. There are two inequivalent embeddings of an n-manifold in the (n+2)-space such that the projection of one of these in (n+1)-space has no double points and the projection of the other has a connected embedded double point set.