Lipschitz geometry of surface germs in R⁴: metric knots

A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in R⁴ is a topological knot (or link) in S³ . We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in R⁴ and knot theory. Namely, for any knot K, we construct a surface X_K in R⁴ such that: the link at the origin of X_K is a trivial knot; the germs X_K are outer bi-Lipschitz equivalent for all K; two germs X_K and XK′ are ambient semialgebraic bi-Lipschitz equivalent only if the knots K and K^′ are isotopic. We show that the Jones polynomial can be used to recognize ambient bi-Lipschitz non-equivalent surface germs in R⁴ , even when they are topologically trivial and outer bi-Lipschitz equivalent.