Robert Tang


Tripper ("Triangle flipper")

Tripper is a Python program for building realisation graphs for systems of curves and arcs on a surface which is currently under development. This is part of a joint project with Piotr Przytycki.

Tripper uses PLink, a graphical interface by Marc Culler and Nathan Dunfield, to draw diagrams of curves and arcs. Tripper outputs the corresponding realisation graph, which can then be visualised using software such as Sage.

The realisation graph for a system of 4 arcs on a disc is an octagon which naturally embeds into a 4-cube.

The realisation graph of a given system of curves and arcs on a surface is defined by setting:

  • Vertices: realisations (or arrangements) of the curves in generic minimal position up to ambient isotopy.
  • Edges: realisations differing by a single "triangle flip", where one curve or arc is pulled across an intersection of two other curves.

Here is a system of arcs on a disc. In this example, each possible realisation corresponds to a Young diagram, which can be used to represent integer partitions.

The corresponding realisation graph is a subgraph of Young's lattice.

Example #1
A system of 5 arcs on a disc drawn using PLink.

The realisation graph of the above system, drawn using Sage (graphviz).

A 3d picture of the same graph drawn in Sage (plot3d using spring layout).

Example #2
A curve and 4 arcs on an annulus.

The realisation graph contains two copies of the "sphere" from Example 1 glued along the magenta/green square in the centre of the picture.