Topological Classification and Finite Determinacy of Knotted Maps

Abstract

We show that the knot type of the link of a real analytic map germ with isolated singularity $f:({\mathbb{R}}^2,0)\to ({\mathbb{R}}^4,0)$ is a complete invariant for $C^0$-$\mathcal{A}$-equivalence. Moreover, we also prove that isolated singularity implies finite $C^0$-determinacy, giving an explicit estimate for its degree. For the general case of real analytic map germs, $f:({\mathbb{R}}^n,0)\to ({\mathbb{R}}^p,0)$ ($n\le p$), we use the Lojasiewicz exponent associated with Mond’s double point ideal $I^2\left(f\right)$ to obtain some criteria of Lipschitz and analytic regularity.