zbMATH — the first resource for mathematics

Reidemeister moves for surface isotopies and their interpretation as moves to movies. (English) Zbl 0808.57020
A movie description of a surface \(F\) embedded in \(\mathbb{R}^ 4\) is a sequence of link diagrams obtained from a projection of \(F\) to \(\mathbb{R}^ 3\) by taking 2-dimensional cross sections perpendicular to a fixed direction on \(\mathbb{R}^ 3\). (In the cross sections, an immersed collection of curves appear, and these are lifted to knot diagrams by using the projection direction from \(\mathbb{R}^ 4\) to \(\mathbb{R}^ 3\).) In this paper, the authors give a set of 15 moves to movies such that two movies represent isotopic surfaces if and only if there is a sequence of moves from this set that takes one to the other. This generalizes the unpublished result of Roseman on projections of surfaces in \(\mathbb{R}^ 4\) to \(\mathbb{R}^ 3\) where a direction of \(\mathbb{R}^ 3\) is not specified. As a warm-up, the authors give a proof of the classical Reidemeister theorem for the case a height function is fixed.
Reviewer: M.Sakuma (Osaka)

57R40 Embeddings in differential topology
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
Full Text: DOI