# Metrically un-knotted corank 1 singularities of surfaces in $\mathbb{R}^4$

## TL;DR

This paper explores the relationship between topological and metric properties of real surface germs in four-dimensional space, showing that normal embedding often implies trivial knotting, with criteria based on polar curves.

## Contribution

It establishes conditions under which normal embedding guarantees trivial knotting of surface links in 4, linking metric and topological properties.

## Key findings

- Normal embedding implies trivial knot for a broad class of surfaces.
- Criteria for normal embedding are provided in terms of polar curves.
- The work connects metric properties with topological triviality of surface links.

## Abstract

The paper is devoted to relations between topological and metric properties of germs of real surfaces, obtained by analytic maps from $R^2$ to $R^4$. We show that for a big class of such surfaces the normal embedding property implies the triviality of the knot, presenting the link of the surfaces. We also present some criteria of normal embedding in terms of the polar curves.