Knotted Surface Diagrams and Their Moves

Knot diagrams and their moves are defined for higher dimensional knots (codimension $2$ smooth embeddings). In particular, for surfaces in $4$-space, they are explicitly known, and depicted in Figs. , , taken from [CKS00]. Specifically, a knotted surface is a smooth embedding $f: F \rightarrow \mathbb{S}^4$ or $\mathbb{R}^4$ and all definitions are given in completely analogous manner as for the classical case. The set of crossing points, in this case, are described by embedded double point curves (A), isolated triple points (B), and branch points (C) in Fig. . The moves analogous to Reidemeister moves, in particular, are obtained by Roseman [Rose98] and called Roseman moves.

Figure: Broken surface diagrams near singular points

Figure: Roseman moves

Referring the reader to [CKS00] for details, we only mention here the relation between knotted surface diagrams and Reidemeister moves for classical knots. The continuous family of projection curves in $\mathbb{R}^2 \times [0,1]$ of a knot, where $[0,1]$ represents the time parameter during which a Reidemeister move occurs, forms a surface mapped in $\mathbb{R}^2 \times [0,1]$. An example for the type I move is depicted in Fig. . As indicated in the figure, the exact moment that the type I move happens corresponds to a branch point of a mapped surface, a generic singularity of smooth maps $F \rightarrow \mathbb{R}^3$ from surfaces to $\mathbb{R}^3$. Similarly, a type II move corresponds to Morse critical points of double point curves of surfaces, and a type III move to an isolated triple point (see also [CrSt98]). Thus a generic singularities and critical points of one dimensional higher case give rise to moves of diagrams. This will be the case for triangulated manifolds as we will see later.

Figure: Continuous family of curves at type I move