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Topological Classification and Finite Determinacy of knotted maps

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