Presentations of 3-manifolds by Heegaard diagrams

Currently the program treats Heegaard diagrams of genus 2, both genuine and extended ones.

Genuine Heegaard diagrams of genus 2

These diagrams are presented as follows:

Heegaard a b c d e f

where a, b, c, d, e, f are positive integers. For example, inputing "Heegaard 4 2 1 3 8 2" yields the following answer:

Stallings manifold with fiber a torus: ( -2 -1 ) ( -1 -1 )	{1 1 } Sol	          H_1=Z_5+Z

The meaning of the notation is indicated in the figure below. The numbers a, b, c, d show the numbers of relevant arcs. The diagram is considered to be drawn on a 3-ball, and the Heegaard diagram is obtained by gluing together the pairs of round discs (in the figure, each of the top discs is glued to the one placed right below it), rotating them as the numbers e and f indicate.

Please notice that the program treats one diagram at a time, namely, if the text in the current window contains more than one Heegaard diagram code, the program takes into account only the first one and ignores the rest.

Extended Heegaard diagrams of genus 2

These diagrams are presented as follows:

ExtHeegaard a b c d e f

where a, b, c, d, e, f are positive integers. For example, inputing "ExHeegaard 9 9 7 8 8 6" yields the following answer:

 Nongeometric	          H_1=Z_2+Z_2
	1	Compound
	Edge 1 { Seifert atom 1 , Seifert atom 2 }	(1 2) (1 1)
Seifert atom 1 (D^2,(2,1),(2,1),(1,0) )
Seifert atom 2 (D^2,(2,1),(3,1),(1,-1) )
	2	Dehn filling Q_3(1,2)

The meaning of the notation is indicated in the figure below. The two collections, each composed of 3 discs and some number of arcs (whose amount is given by numbers a, b, c), are drawn on two distinct 3-balls. These balls are then glued together along the pairs of round discs (joined by the arrows in the figure). The numbers d, e, f indicate how the discs should be rotated.

Once again we mention that the program treats one diagram at a time, namely, if the text in the current window contains more than one diagram code, the program takes into account only the first one and ignores the rest.

Retrievieing information on Heegaard diagrams

If you are using the presentation by Heegaard diagrams, you can view some additional information about it. Namely, if you go to "Information|Short", you can view the following items: