Tutorial on hyperbolic surfaces

A hyperbolic surface with cusps is a topological surface together with a hyperbolic structure of finite area. polymake can deal with hyperbolic surfaces in view of Penners coordinates of the decorated Teichmüller space (lambda lengths). These allow to pick a hyperbolic surface by choosing a triangulation of the surface along with one positive parameter for each edge.

The secondary fan of a hyperbolic surface stratifies the space of weight vectors (horocyclic decorations) according to which Delaunay triangulations are induced by the Epstein-Penner convex hull construction. For each point on the surface, there is a secondary polyhedron whose normal fan is the secondary fan.

This tutorial shows how to deal with secondary fans and secondary polyhedra of hyperbolic surfaces.

To define a hyperbolic surface we need to specify (a) a triangulation and (b) Penner coordinates.

(a) The triangulation is obtained by specifying the DCEL_DATA as an Array<Array<Int». This constructs a doubly connected edge list as follows: Each row of DCEL_DATA reads { (2i).head , (2i+1).head , (2i).next , (2i+1).next }. In general, for each edge i of the triangulation there are two half edges 2i and 2i+1, one for each orientation.

(b) The PENNER_COORDINATES assign a positive rational number to each edge of the triangulation, ordered in the same sense as prescribed by the DCEL_DATA.

> application 'fan';
> application 'topaz';
> $S3 = new Array<Array<Int>>([[1,0,2,5],[2,1,4,1],[0,2,0,3]]);
> $s = new HyperbolicSurface(DCEL_DATA=>$S3,PENNER_COORDINATES=>[1,1,1]);

In this example the doubly connected edge list looks like this:

The secondary fan of the hyperbolic sphere from above can now be computed as follows.

> $f = $s->SECONDARY_FAN;
> $f->properties;
type: PolyhedralFan<Rational>
 
RAYS
0 1 1
1 0 1
1 1 0
0 0 1
1 0 0
0 1 0
 
 
MAXIMAL_CONES
{0 1 2}
{0 1 3}
{1 2 4}
{0 2 5}
 
> $f->VISUAL;
> $s->properties;
name: s
type: HyperbolicSurface
 
DCEL_DATA
1 0 2 5
2 1 4 1
0 2 0 3
 
 
PENNER_COORDINATES
1 1 1
 
SECONDARY_FAN
type: PolyhedralFan<Rational>
 
FLIP_WORDS
{}
{0}
{1}
{2}

The FLIP_WORDS indicate how to obtain the Delaunay triangulations. The k-th flip word is a list of integers (the indices of the edges) that describe which edge flips produce the k-th Delaunay triangulation. Note that the k-th Delaunay triangulation also corresponds to the k-th maximal cone of the SECONDARY_FAN.

In order to compute GKZ_VECTORS or a secondary_polyhedron of a hyperbolic surface one needs to additionally specify a SPECIAL_POINT on the surface. This is done by choosing two rational numbers.

Continuing with the above example, lets look at the following.

> $s = new HyperbolicSurface(DCEL_DATA=>$S3,PENNER_COORDINATES=>[1,1,1],SPECIAL_POINT=>[1,0]);

Now we may compute an approximation of the GKZ_VECTORS of the surface. The approximation depends on a parameter depth that restricts the depth of the (covering) triangles that are summed over in the definition of the GKZ vectors.

> print $s->GKZ_VECTORS(3);
1 33346854621/25672050625 33346854621/25672050625 19782163/27238250
1 2361/3250 3955357/5447650 33346854621/25672050625
1 10549213550005124385885122/6365327663846199230365625 11433978/13287625 30327974429709/105771923977850
1 11433978/13287625 10549213550005124385885122/6365327663846199230365625 30327974429709/105771923977850

The secondary polyhedron can be computed similarly using the function secondary_polyhedron.

> $p = secondary_polyhedron($s,10);
> $p->properties;
name: p
type: Polytope<Float>
 
VERTICES
1 1.315301353 1.315301353 0.7316378744
1 0.7316489581 0.7316267908 1.315301353
1 1.752046187 0.8750928112 0.2910011302
1 0.8750928112 1.752046187 0.2910011302
0 -1 0 0
0 0 -1 0
0 0 0 -1
 
 
VERTICES_IN_FACETS
{0 1 3 4}
{0 1 2 5}
{0 2 3 6}
{1 4 5}
{2 5 6}
{3 4 6}
 
 
CONE_AMBIENT_DIM
4
> $p->VISUAL(FacetColor=>'255 180 80');

We may look at the GKZ domes of the individual Delaunay triangulations.

> $d0 = $s->gkz_dome(0,5);
> $d0->VISUAL(FacetColor=>'80 180 255');
> $d1 = $s->gkz_dome(1,5);
> $d1->VISUAL(FacetColor=>'80 180 255');

> $T3 = new Array<Array<Int>>([[1,0,2,17],[2,1,4,14],[0,2,0,6],[1,2,8,16],[0,1,5,10],[2,1,12,1],[0,2,9,3],[0,1,13,7],[0,2,15,11]]);
> $s = new HyperbolicSurface(DCEL_DATA=>$T3, PENNER_COORDINATES=>[2,1,1,1,1,1,1,1,1], SPECIAL_POINT=>[1,0]);

> $f = $s->SECONDARY_FAN;
> $f->VISUAL;
> $s->properties;
name: s
type: HyperbolicSurface
 
DCEL_DATA
1 0 2 17
2 1 4 14
0 2 0 6
1 2 8 16
0 1 5 10
2 1 12 1
0 2 9 3
0 1 13 7
0 2 15 11
 
 
PENNER_COORDINATES
2 1 1 1 1 1 1 1 1
 
SPECIAL_POINT
1 0
 
SECONDARY_FAN
type: PolyhedralFan<Rational>
 
FLIP_WORDS
{0}
{}
{0 3}
{0 4 7}
{0 6}
{3}
{6}
{0 3 1 5}
{0 6 2 8}
{3 1 5}
{6 2 8}
{0 3 1 5 0 1}
{0 6 2 8 0 2}
 
> $p = secondary_polyhedron($s,7);
> $p->VISUAL(FacetColor=>'255 180 80');
> $d0 = $s->gkz_dome(0,5);
> $d0->VISUAL(FacetColor=>'80 180 255');
> $s = new HyperbolicSurface(DCEL_DATA=>$T3, PENNER_COORDINATES=>[2,1,1,1,1,1,1,1,1], SPECIAL_POINT=>[new Rational(1.5196714),new Rational(-0.5773503)]);
> $p = secondary_polyhedron($s,7);
> $p->VISUAL(FacetColor=>'255 180 80');
> $d0 = $s->gkz_dome(0,5);
> $d0->VISUAL(FacetColor=>'80 180 255');

> # a torus with two cusps (6 edges)
> $T2 = new Array<Array<Int>>([[0,0,6,5],[0,0,1,10],[0,0,8,2],[1,0,11,4],[1,0,7,3],[1,0,9,0]]);
> 
> # a sphere with four cusps (6 edges)
> $S4 = new Array<Array<Int>>([[1,0,2,6],[2,1,4,9],[0,2,0,11],[3,0,8,5],[1,3,1,10],[2,3,3,7]]);
> 
> # a double torus with two cusps (12 edges)
> $DT2 = new Array<Array<Int>>([[0,0,8,10],[0,0,12,14],[0,0,16,18],[0,0,20,22],[1,0,23,2],[1,0,13,3],[1,0,9,1],[1,0,11,4],[1,0,15,6],[1,0,21,7],[1,0,17,5],[1,0,19,0]]);

To study 4-dim. secondary fans the following method is useful. It intersects the secondary fan with the 3-dim. standard simplex.

> sub norm($){
>    my $B = new Matrix(shift);
>    for (my $i = 0; $i < $B->rows(); ++$i) {
>       my $sum = 0;
>       for (my $j = 1; $j < $B->cols(); ++$j) {
>          $sum = $sum + $B->elem($i,$j);
>       }
>       $x = 1/$sum;
>       $B->row($i) = $x * $B->row($i);
>    }
>    return $B;
> }
> $s = new HyperbolicSurface(DCEL_DATA=>$S4,PENNER_COORDINATES=>[1,1,1,1,1,1],SPECIAL_POINT=>[1,0]);
> $f = $s->SECONDARY_FAN;
> $v = ones_vector | $f->RAYS;
> $a = norm($v);
> $b = $a->minor(All,~[0]);
> $c = ones_vector | $b;
> $q = new fan::PolyhedralComplex(POINTS=>$c,INPUT_POLYTOPES=>rows($f->MAXIMAL_CONES));
> $pro = fan::project_full($q);
> $pro->VISUAL;