For a real analytic map germ from the plane to four-space with isolated singularity, intersecting the image with a small sphere produces a knot in the 3-sphere, and the knot type of that link is a complete invariant: two such germs are equivalent under homeomorphisms of source and target exactly when their links are equivalent knots. A student learns how invertible cobordism from both ends, combined with the classification of knots by knot group plus peripheral structure, forces topologically equivalent germs to have equivalent links. The same setting yields Lojasiewicz exponents of the double point ideal that bound the Taylor order at which a germ's topological type is already determined, and that detect when a bi-Lipschitz parametrization is in fact a smooth embedding.
A map germ is a real analytic map f:(R^n,0)->(R^p,0) studied only near the origin; it has isolated singularity when some representative is a topological embedding that is an immersion away from 0, so…