Surface singularities in $R^4$: first steps towards Lipschitz knot theory

A link of an isolated singularity of a two-dimensional semialgebraic surface in $R^4$ is a knot (or a link) in $S^3$. Thus the ambient Lipschitz classification of surface singularities in $R^4$ can be interpreted as a bi-Lipschitz refinement of the topological classification of knots (or links) in $S^3$. We show that, given a knot $K$ in $S^3$, there are infinitely many distinct ambient Lipschitz equivalence classes of outer metric Lipschitz equivalent singularities in $R^4$ with the links topologically equivalent to $K$.