Applicable Moves
Generally, there are the following moves that the program may
apply to a given spine:
- cell reductions. A typical situation is when there is an edge adjacent
to exactly two cell germs, and the germs belong to distinct cells;
- cell collapses;
- rough piercing. This move is applied when there is a boundary component
that is a 2-sphere. The effect of the move will be to transform this boundary component
into a torus;
- delicate piercing. This is applied when there are at least two boundary
components, one of which is a sphere;
- a short curve reduction. This is relevant if the spine contains a 2-cell whose
closure (in the spine) is an embedded disc. This move yields a spine of the same
manifold, and, if the boundary of the cell passes through no more
than three vertices, this move allows to obtain a new spine with smaller
number of vertices. (Do note that the new spine may not be special. But if
the cell passes through exactly three vertices, it is). If it passes through
four vertices, then (in general) the new spine will have the same number of vertices,
but it is different and may be better for further simplification;
- surgeries: cutting a free strip and cutting along an annulus;
- counterpass simplification;
- replacement with a short component.
- cutting along normal torus.
A precise description of moves should be seen in the literature (see, for instance,
[Matveev]).
[Matveev] S. Matveev, Algorithmic topology and classification of 3-manifolds,
Springer, 2003.