Spherical algebraic knots increase with dimension
Let \(f(z_ 0,\dots,z_ n)\) be a polynomial in complex variables, having an isolated singularity at the origin. If the zero set \(f^{-1}(0)\) intersects a small sphere \(S^{2n+1}_ \varepsilon\) in a manifold \(\Sigma\) homeomorphic to a sphere, then \(\Sigma\) is called a spherical algebraic knot. Write \(A_ n\) for the set of isomorphism classes of such knots \(\Sigma\). Replacing \(f\) by \(f+z^ 2_{n+1}+z^ 2_{n+2}\) induces an injective map \(A_ n\to A_{n+2}\). By adapting an example of Malgrange, the authors construct a knot in \(A_{n+1}\) whose monodromy has a Jordan block of size \(n\). Such a knot is not in the image of \(A_{n-3}\).
scientific article; zbMATH DE number 124279