Lipschitz geometry of surface germs in $${\mathbb {R}}^4$$: metric knots

Abstract

AbstractA link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in $${\mathbb {R}}^4$$ R 4 is a topological knot (or link) in $$S^3$$ S 3 . We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in $${\mathbb {R}}^4$$ R 4 and knot theory. Namely, for any knot K, we construct a surface $$X_K$$ X K in $${\mathbb {R}}^4$$ R 4 such that: the link at the origin of $$X_{K}$$ X K is a trivial knot; the germs $$X_K$$ X K are outer bi-Lipschitz equivalent for all K; two germs $$X_{K}$$ X K and $$X_{K'}$$ X K ′ are ambient semialgebraic bi-Lipschitz equivalent only if the knots K and $$K'$$ K ′ are isotopic. We show that the Jones polynomial can be used to recognize ambient bi-Lipschitz non-equivalent surface germs in $${\mathbb {R}}^4$$ R 4 , even when they are topologically trivial and outer bi-Lipschitz equivalent.