Abstract
A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in {mathbb {R}}^4 is a topological knot (or link) in S^3. We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in {mathbb {R}}^4 and knot theory. Namely, for any knot K, we construct a surface X_K in {mathbb {R}}^4 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 X_{K'} 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 {mathbb {R}}^4, even when they are topologically trivial and outer bi-Lipschitz equivalent.
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