Intro to (Algebraic) Topology Algebraic topology. Computational topology: an introduction

use
__Method -> Effect -> Explode Group of Geometries__ in the JavaView menu.
Options:
__mixed_graph__ => Bool use the __MIXED_GRAPH__ for the spring embedder
__seed__ => Int random seed value for the string embedder
Options: Attributes modifying the appearance of filled polygons.
__FacetColor__ => Color filling color of the polygon
__FacetTransparency__ => Float transparency factor of the polygon between 0 (opaque) and 1 (completely translucent)
__FacetStyle__ => String if set to "hidden", the inner area of the polygon is not rendered
__EdgeColor__ => Color color of the boundary lines
__EdgeThickness__ => Float scaling factor for the thickness of the boundary lines
__EdgeStyle__ => String if set to "hidden", the boundary lines are not rendered
__Title__ => String the name of the drawing
__Name__ => String the name of this visual object in the drawing
__Hidden__ => Bool if set to true, the visual object is not rendered
(useful for interactive visualization programs allowing for switching details on and off)
__PointLabels__ => String if set to "hidden", no point labels are displayed
__VertexLabels__ => String alias for PointLabels
__PointColor__ => Flexible color of the spheres or rectangles representing the points
__VertexColor__ => Flexible alias for PointColor
__PointThickness__ => Flexible scaling factor for the size of the spheres or rectangles representing the points
__VertexThickness__ => Flexible alias for PointThickness
__PointBorderColor__ => Flexible color of the border line of rectangles representing the points
__VertexBorderColor__ => Flexible alias for PointBorderColor
__PointBorderThickness__ => Flexible scaling factor for the thickness of the border line of rectangles representing the points
__VertexBorderThickness__ => Flexible alias for PointBorderThickness
__PointStyle__ => Flexible if set to "hidden", neither point nor its label is rendered
__VertexStyle__ => Flexible alias for PointStyle
__ViewPoint__ => Vector ViewPoint for Sketch visualization
__ViewDirection__ => Vector ViewDirection for Sketch visualization
__ViewUp__ => Vector ViewUp for Sketch visualization
__Scale__ => Float scale for Sketch visualization
__LabelAlignment__ => Flexible Defines the alignment of the vertex labels: left, right or center
Options: Attributes modifying the appearance of graphs
__Coord__ => Matrix 2-d or 3-d coordinates of the nodes.
If not specified, a random embedding is generated using a pseudo-physical spring model
__NodeColor__ => Flexible alias for PointColor
__NodeThickness__ => Flexible alias for PointThickness
__NodeBorderColor__ => Flexible alias for PointBorderColor
__NodeBorderThickness__ => Flexible alias for PointBorderThickness
__NodeStyle__ => Flexible alias for PointStyle
__NodeLabels__ => String alias for PointLabels
__ArrowStyle__ => Flexible How to draw directed edges: 0 (like undirected), 1 (with an arrow pointing towards the edge),
or -1 (with an arrow pointing against the edge). Default is 1 for directed graphs and lattices.
__EdgeColor__ => Flexible color of the lines representing the edges
__EdgeThickness__ => Flexible scaling factor for the thickness of the lines representing the edges
__EdgeLabels__ => EdgeMap textual labels to be placed along the edges
__EdgeStyle__ => Flexible if set to "hidden", neither the edge nor its label is rendered
__Title__ => String the name of the drawing
__Name__ => String the name of this visual object in the drawing
__Hidden__ => Bool if set to true, the visual object is not rendered
(useful for interactive visualization programs allowing for switching details on and off)
__PointLabels__ => String if set to "hidden", no point labels are displayed
__VertexLabels__ => String alias for PointLabels
__PointColor__ => Flexible color of the spheres or rectangles representing the points
__VertexColor__ => Flexible alias for PointColor
__PointThickness__ => Flexible scaling factor for the size of the spheres or rectangles representing the points
__VertexThickness__ => Flexible alias for PointThickness
__PointBorderColor__ => Flexible color of the border line of rectangles representing the points
__VertexBorderColor__ => Flexible alias for PointBorderColor
__PointBorderThickness__ => Flexible scaling factor for the thickness of the border line of rectangles representing the points
__VertexBorderThickness__ => Flexible alias for PointBorderThickness
__PointStyle__ => Flexible if set to "hidden", neither point nor its label is rendered
__VertexStyle__ => Flexible alias for PointStyle
__ViewPoint__ => Vector ViewPoint for Sketch visualization
__ViewDirection__ => Vector ViewDirection for Sketch visualization
__ViewUp__ => Vector ViewUp for Sketch visualization
__Scale__ => Float scale for Sketch visualization
__LabelAlignment__ => Flexible Defines the alignment of the vertex labels: left, right or center
Returns Visual::SimplicialComplex

for a list of available options and this tutorial for a general intro to visualization in polymake.
If your complex is of dimension three or lower, you can visualize a geometric realization together with the GRAPH of the complex using the VISUAL property. Note that if your complex is not a GeometricSimplicialComplex, polymake will use the spring embedder to find an embedding of the graph of the complex, which is not guaranteed to result in an intersection-free visualization.
> $bs->VISUAL;
You should give the explode feature of jReality a try – it gives a good (and pretty!) overview of the object. You can find it in the left slot of the jReality interface.
topaz may also visualize distinguished subcomplexes or just sets of faces with different decorations (colors, styles, etc.). For example, to highlight the fourth facet of $bs in pink, do this:
> $a = new Array>(1); $a->[0] = $bs->FACETS->[4];
> $bs->VISUAL->FACES($a, FacetColor => 'pink');
The same can be used for the visualization of the face lattice. As an example, we have a look at a morse matching of the Klein bottle with its associated critical faces. In order to see the arrowheads in the picture clearly, you ought to use graphviz or svg to vizualize it.
> $k = klein_bottle();
> svg($k->VISUAL_FACE_LATTICE->MORSE_MATCHING->FACES($k->MORSE_MATCHING->CRITICAL_FACES));

Here the matching of faces is denoted by reversed red arrows and the critical faces are marked red. Check that the graph remains acyclic.
For higher dimensional complexes that cannot be visualized in 3D, you can still have a look at the graphs while ignoring any specified coordinates by using VISUAL_GRAPH, VISUAL_DUAL_GRAPH, or VISUAL_MIXED_GRAPH. An easy example:
> polytope::cube(3)->TRIANGULATION->VISUAL_MIXED_GRAPH;

shows the primal and dual graph of the polytope together with an edge between a primal and a dual node iff the primal node represents a vertex of the corresponding facet of the dual node.

Visualization of the HASSE_DIAGRAM is possible via VISUAL_FACE_LATTICE. It renders the graph in a .pdf file. You can even pipe the tikz code to whatever location using the tikz client:
tikz($s->VISUAL_FACE_LATTICE, File=>"/path/to/file.tikz");