Lipschitz geometry of surface germs in R4: metric knots
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Lev Birbrair [1] ; Michael Brandenbursky [3] ; Andrei Gabrielov [2]
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[1] Universidade Federal do Ceará
Universidade Federal do Ceará
Brasil
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[2] Purdue University
Purdue University
Township of Wabash, Estados Unidos
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[3] Ben Gurion University of the Negev, Israel
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- Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 29, Nº. 3, 2023
- Idioma: inglés
- Enlaces
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Resumen
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.
