Topology and Lipschitz regularity of algebraically parametrized surfaces in $\mathbb{R}^4$

It is proved by Pham and Teissier in [PT] (also Fernandes in [F]) that two irreducible complex plane curve singularities in $\mathbb{R}^4$ are outer bi-Lipschitz equivalent if and only if are topologically equivalent. This result was generalized by Pichon and Neumann in [NP]. The topological type, in this case, is equivalent to the knot type of your "link" (that is always an iterated knot). In this talk, we consider the similar approach for the general case, that is, singular real surfaces $X$ in $\mathbb{R}^4$ parametrized by polynomial map germs $f \colon (\mathbb{R}^2,0){\rightarrow }(\mathbb{R}^4,0)$ with isolated singularity. We show that, given $X{=}f(\mathbb{R}^2)$, the knot type of the link $X \cap \mathbb{S} ^3(0,\epsilon)$ determines completely the $C^0$-$\mathscr{A}$-class of $f$ and all parametrizations of this type are $C^0$-finitely determined. Moreover, we show that if $X$ is a bi-Lipschitz embedded parametrized surface, then $X$ is smooth. This is a joint work with Juan Jose Nuno Ballesteros. [BFLS] Birbrair L., Fernandes A., Lê, D. T., Sampaio J. E., {\it Lipschitz regular complex algebraic sets are smooth}, Proc. Amer. Math. Soc. 144 (2016), no. 3, 983--987. [F] Fernandes A., {\it Topological equivalence of complex curves and bi-Lipschitz homeomorphisms}, Michigan Math. J. 51 (2003), n. 3, 593--606. [NP] Neumann W. D., Pichon, A., {\it Lipschitz geometry of complex curves}, J. Singul. 10 (2014), 225--234. [PT] Teissier B., Pham, F, {\it Fractions lipschitziennes d'une alg\'ebre analytique complexe et saturation de Zariski}, Centre de Math\'ematiques de l'Ecole Polytechnique (Paris), June 1969.Non UBCUnreviewedAuthor affiliation: UNILAB-CearáFacult