THREE-MANIFOLD RECOGNIZER
INSTALLING THE RECOGNIZER
It is necessary only to copy Recognizer.exe to some directory. In order to use this
help more efficiently, it is also
desirable to create in the same directory a subdeirectory named "Help" and put the
present help-file in there. This will allow to use the Help directly from the
program (either by pressing F1 or referring the menu "Help").
USING THE RECOGNIZER
What the Recognizer can do:
- Recognize some types of manifolds (closed or with boundary). What, any manifold?
- From a given presentation of a manifold, give the following information:
- first homology group;
- values of Turaev-Viro invariants up to order 16;
- values of Epsilon-invariants up to order 16;
- value of Casson-Walker Invariant if surgery presentation of the manifold are used
(now surgeries along knots and 2-component links only);
- its geometric type (after the complete recognition);
- hyperbolic volume, if the manifold is hyperbolic and closed.
- From a given spine, give the following information:
- whether it is thickenable;
- the induced cell decomposition of the boundary of
its regular neighborhood;
How does it look like?
- the code of the singular graph;
How does it look like?
- the list of all closed surfaces embedded into the spine;
- the list of normal surfaces;
- the list of symmetries of the spine (notice that each symmetry is described
by its action on the boundary, so in particular for closed manifolds the only useful
information provided will be the number of symmetries).
- Perform individual elementary transformations of a given special spine
(and the corresponding manifold). Which ones are available?
- Retrieve a presenation of the special spine with which the program is currently
working (this is possible even if you started with some other way of presenting the
manifold, because the program always starts by building some special spine of it).
This can be quite useful if you've just availed of the opportunity mentioned in the
previous item and want to observe the effect on the spine.
Accepted presentations of manifolds:
(please refer to the menu "Help|Examples" for information about this presentations).
Accepted presentations of spines:
There are two presentations available.
How to ask for what you want:
The starting point is obviously to describe a manifold, by either
going to "File|New" in the main menu and entering a presentation of
one of available types, or by going to "File|Open" and opening a file with
such a presenation prepared in advance. Then you may proceed as
follows:
- to request one of the information items listed above, choose a
relevant item from "Information" menu;
- to recognize an individual manifold, go to "Recognizer|Recognize" and wait;
- to perform an elementary move on the spine, go to "Recognize|Dialog" and
choose one of the proposed options;
- to observe the recognition process one step at a time, go to "Recognizer|Make One Step".
Only one step is performed and you can observe the effect on the spine by going to the
information menu. At this point you can also "improve" the cellular decomposition of your spine
by going to "Recognizer|Make Cell Reductions";
- to retrieve a presentation of the special spine with which
the program is working, go to "Information|Show Boundary Curves"
(to get a presentation by boundary curves) or to "Information|Show Details"
(to get a presentation by gluing of details), see
the explanation. If you are in a dialog,
you'll need to close it first.
- To simplify a special spine go "Recognizer|Simplify Special Spine". It is possible for any of available presentations.
Note that for performing the recognition some
advanced options are available.
What do the results mean?
- Results of recognition:
The result of program's work is given by a labelled molecule.
Informally, that is a graph whose vertices are some manifolds with torus
boundary components and the edges correspond to gluings between these tori.
Here is an example which is obtained by running the Recognizer
(with default settings) on the following spine. In simple cases
(such as when a manifold is geometric), you do not really need to understand
the labelled molecule.
The labelled molecule is followed by the result of so-called assembling,
which is an attempt to produce the JSJ-decomposition of the manifold in
question. Assume first that the program has been able to produce a labelled
molecule without exceptional vertices. Then there are several posibilities:
- The manifold is Seifert. Then the program outputs the Seifert structure
in the usual format, by indicating the base surface and the parameters of
exceptional fibers, see example. The answer
also includes the geometric type of the manifold and its homology group;
- The manifold is hyperbolic. Here is an example
of running the Recognizer on a special spine
of a hyperbolic manifold. Notice that the fact that the result is presented as a Dehn
filling does not guarantee that the manifold is hyperbolic; in order to check that,
one should attempt to calculate the volume by going to
"Information|Hyperbolic Volume".
- The manifold has a non-trivial JSJ-decomposition. Here is an
example
of running the Recognizer on a special spine
of a manifold with a nontrivial JSJ-decomposition.
It may happen that the labelled molecule contains some exceptional vertices. Then
the output of the program is a "quasi JSJ-decomposition", i.e. the program
describes a manifold as a result of gluing along tori some Seifert manifolds, some
hyperbolic pieces, and some unknown ones.
- Results of spine transformations:
The program outputs the spine in either of the two forms in which it
receives it, see the explanation. For instance,
one may ask the program to apply the inverse T-move to
this spine, which contains an embedded
component of length 3, by going to "Recognizer|Dialog" and choosing the transformation
"Short curve" (notice that there are generally several possible ways to
apply this move; the specific cell to which the chosen move applies is indicated
in the preceding line as "Crude piercing 1 = { 4, 2, 7 }"),
and view the result in the following
form.
Go back and do something new:
Initial presentation can always be retrieved by going to "Recognizer|Initial State".
If you had only viewed various information items, you can also return to the
initial presentation by going to "Information|Remove Information".
Quick help
For quick reference, the menu "Help|Examples" allows to see concrete examples
for the typical presentations of manifolds listed above.
"Three-manifold Recognizer"
is being developed by the research group of S. Matveev in the department
of computer topology and algebra of Chelyabinsk State University.
Contacts