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

In this paper we are concerned with labelled apparent contours, namely with apparent contours of generic orthogonal projections of embedded surfaces in R3, endowed with a suitable depth information. We show that there exists a finite set of elementary moves (i.e. local topological changes) on labelled apparent contours such that the following theorem holds: two generic embeddings of a closed surface S in R3 are isotopic if and only if their apparent contours can be connected using only smooth isotopies and a finite sequence of moves.