Special spine presentations of 3-manifolds

Presentation by boundary curves

Assume that you have a special spine P. In order to write down the presentation accepted by the program, do the following:

  1. Number the edges of P and orient them;
  2. For each 2-cell choose a basepoint and an orientation on its boundary curve. Then write down a string of integers which shows in what order and in which direction the boundary curve of the cell passes along the edges.

Example.

Suppose that P is the spine shown in the figure below. It can be constructed as follows. Take the graph with two vertices and four oriented edges a,b,c,d, where a is a loop with ends in one vertex, d is a loop is a loop with ends in the other vertex, and b,c join the vertices. (The orientations of b,c are such that they go out of the first vertex.) Glue three 2-cells along the curves abc^{-1}, aacddb^{-1}, bdc^{-1}. (This CW complex is actually a special spine of the lens space L(5,1), you can look it up in the table of manifolds of complexity 2 given in [Matveev]). If we number edges a,b,c,d respectively with 1,2,3,4 and keep the orientations, then Steps 1 and 2 above produce the following presentation:

                          1 2 -3
                          1 1 3 4 4 -2
                          2 4 -3  

Remarks.

  1. The minus sign shows that the boundary curve passes along the edge in the direction opposite to the orientation of the edge.
  2. Strings corresponding to different cells should be written in different lines.
  3. The integers in each string should be separated by one or more spaces.
  4. There are some other styles to write a presentation of 2-cell boundary curve, for example any curve can have a number or a name. These styles can be viewed in more detail by going to "Help|Examples|Three styles of presenting special spines"

Presentation by gluing details

This is an alternative way of presenting special spines. It is more complicated but it gives some idea of how the neighborhood of the singular graph looks like. For example, the presentation by gluing details for the above spine is:

v1 =
	1 : 	v1	_2	 { 1 3 2}	% *** Edge 1 beginning
 %	2 : 	v1	_1	 { 1 3 2}	% *** Edge 1 end
	3 : 	v2	_1	 { 3 1 2}	% *** Edge 2 beginning
	4 : 	v2	_2	 { 3 1 2}	% *** Edge 3 beginning
v2 =
 %	1 : 	v1	_3	 { 2 3 1}	% *** Edge 2 end
 %	2 : 	v1	_4	 { 2 3 1}	% *** Edge 3 end
	3 : 	v2	_4	 { 3 2 1}	% *** Edge 4 beginning
 %	4 : 	v2	_3	 { 3 2 1}	% *** Edge 4 end

Here v1 and v2 are vertices of the spine, the numbers 1,2,3,4 in the first column of each vertex's description denote the four edge germs at each vertex, and the symbol v1 or v2 at the next column shows where the corresponding edge goes if we leave out that germ. The triple of form {1,3,2} describes the way the triods (comprised of germs of cells) are glued when the germs of edges are glued. The symbol % stands for the beginning of an one-line commentary, hence whatever follows it in the line is not necessary for the program.