Morse Matching Example
We provide an example of a so-called Morse matching on a
simplicial complex formed by the letters
"BAT". See the description below for details. The
visualization is performed by
. "BAT" stands for "Berliner
Algorithmen Tag" ("Berlin Algorithm
Day"). I presented this example in my talk for the
|Improve the drawing by pressing "Shift-z" ...
drag to rotate
press "w" to auto-rotate ("q"
press "s" and drag to scale
press "t" and drag to translate
press mouse-right to show menu
The above applet visualizes a so-called Morse Matching
on the simplicial complex of the three letters
"BAT" using JavaView .
The simplicial complex consists of the collection of
triangles, edges, and vertices you see in the picture. The
elements of the simplicial complex are called faces. The
Morse Matching is a matching of vertices to edges and
edges to triangles such that each element of the complex is
matched with at most one other element. Furthermore there is no
directed cycle in the pictures, i.e., when you follow the arrows
you cannot come back to the face you started from.
Morse matchings capture the main structure of discrete
Morse Functions which were introduced by Robin Forman in
1998 . The concept of Morse matchings was introduced by Manoj
Chari . For more details on computing Morse matchings with as
few critical faces as possible see reference .
In the example, the faces which are not matched are called
critical and are shown in red. (Unfortunately the
critical vertices cannot be visualized with the current JavaView
- For the "T" there are one critical triangle and
one critical vertex. Topologically, the "T" is
(homotopy equivalent to) a sphere. Indeed, there is a theorem
by Robin Forman  which states that a combinatorial manifold
(without boundary) with two critical faces is homeomorphic to
- The "A" has one critical triangle, two critical
edges, and one critical vertex. It can be deformed to a
- Finally, the "B" contains one critical triangle,
four critical edges, and one critical vertex. You can see that
it can be deformed to a double-torus.
For more information see the following references.
Manoj K. Chari, On discrete Morse functions and combinatorial decompositions
Discrete Math. 217 (2000), No. 1-3, pp. 101-113.
Robin Forman, Morse Theory for Cell-Complexes
Advances in Math. 134 (1998), pp. 90-145.
Robin Forman, Combinatorial Differential Topology and Geometry
New Perspectives in Geometric
Combinatorics, L. Billera et al. (eds.), Cambridge
University Press, Math. Sci. Res. Inst. Publ. 38, 177-206
Michael Joswig and Marc E. Pfetsch,
Computing Optimal Morse Matchings
SIAM J. Discrete Math. 20, no. 11 (2006), 11-25
Konrad Polthier et al. JavaView