Resumen de Lipschitz geometry of surface germs in R4: metric knots

Lev Birbrair, Michael Brandenbursky, Andrei Gabrielov

• 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 XK in R4 such that: the link at the origin of XK is a trivial knot; the germs XK are outer bi-Lipschitz equivalent for all K; two germs XK 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.