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

Abstract: A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in $\R^4$ is a topological knot (or link) in $S^3$. We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in $\R^4$ and the knot theory. Namely, for any knot $K$, we construct a surface $X_K$ in $\R^4$ such that: $X_K$ has a trivial knot at the origin; the germs $X_K$ are outer bi-Lipschitz equivalent for all $K$; two germs $X_{K}$ and $X_{K'}$ are ambient bi-Lipschitz equivalent only if the knots $K$ and $K'$ are isotopic.

differential geometrydynamical systemsgeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and Dynamics seminar

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