# Completeness of Reidemeister-type moves for surfaces embedded in three-dimensional space

### Giovanni Bellettini

Università di Roma 'Tor Vergata', Italy### Valentina Beorchia

Università di Trieste, Italy### Maurizio Paolini

Università Cattolica del Sacro Cuore, Brescia, Italy

## Abstract

In this paper we are concerned with labelled apparent contours, namely with apparent contours of generic orthogonal projections of embedded surfaces in $\mathbb R^3$, endowed with a suitable information on the relative depth. We give a proof of the following theorem: there exists a finite set of elementary moves (i.e. local topological changes) on labelled apparent contours such that two generic embeddings in $\mathbb R^3$ of a closed surface are isotopic if and only if their apparent contours can be connected using only smooth planar isotopies and a finite sequence of moves. This result, that can be obtained as a by-product of general results on knotted surfaces and singularity theory, is obtained here with a direct and rather elementary proof.

## Cite this article

Giovanni Bellettini, Valentina Beorchia, Maurizio Paolini, Completeness of Reidemeister-type moves for surfaces embedded in three-dimensional space. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 23 (2012), no. 1, pp. 69–87

DOI 10.4171/RLM/617