A Heegaard diagram for the Poincare homology sphere

The following is a Heegaard diagram for the Poincare homoloogy sphere, viewed as the +1 surgery on the right handed treoil knotl.
In the picutre, the left two small circles correspond to a handle, and so are the right two small circles. So this is a genus two Heegaard diagram. By cutting the Heegaard surface along the two alpha curves, we get the following picture.
We use this picture since all regions are actually connected, and we only have alpha and beta curves. Note that we REQUIRE the following two conditions:
  1. The Heegaard diagrams should be admissible. (This is always true if the three-manifold is an integer homology sphere).
  2. The Heegaard diagram contains no non-disk region.

To input our Heegaard diagram into mathematica, we label all regions (edges) from one to the number of regions (edges) satisfying the following two conditions:
  1. The first region is where we have the basepoint.
  2. For a knot, the second region is the one containing the other marked point.
Now we read the edges of a region counterclockwise, so that the first edge of each disk is an alpha edge. For the Heegaard diagram on the left, we read
  sigma235={
    (*D1 *) {-1, -20, 7, -19, -9, 26},
    (*D2 *) {-10, 19, 8, -18, 1, 14},
    (*D3 *) {-5, -21, 6, 20},
    (*D4 *) {-6, -22, 13, 21, -4, 24},
    (*D5 *) {-13, -23, 12, 22},
    (*D6 *) {-12, -15, 3, -16, 11, 23},
    (*D7 *) {-11, -14, 2, 15},
    (*D8 *) {-3, -25, -7, -24},
    (*D9 *) {-2, -26, -8, 25},
    (*D10*) {4, 17, 10, 16},
    (*D11*) {5, 18, 9, -17}
  }
The program will have some methods to check the correctness of a Heegaard diagram. But please double check to make sure the diagram is what you want.

J. J. 2006.08.30