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 , 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 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