Interactive
Graph rotation systems
An interactive explorer for rotation systems in topological graph theory — edit a graph's local rotations, generate the resulting facial walks dart by dart, and watch the fundamental polygons take shape. The full interaction is built out here for $K_{3,3}$, $K_5$, $K_5 \setminus e$, and the octahedral graph.
Choose a graph
Pick a graph to edit its rotation system and generate its facial walks.
1Rotation system — $K_{3,3}$
Local rotations σv
Each row is a vertex's cyclic edge order. Tap ↻ to cycle through its rotations; the count shows which one of the total.
Note: each graph starts in its drawing's angular rotation, so the embedding shown isn't necessarily the minimum-genus one. The numbers on each edge give that vertex's exact cyclic order; the curved arrow shows its orientation. Cycle the local rotations (↻) to explore other embeddings.
2Facial walks
3Fundamental polygons
Each face opens out into a polygon whose sides are its facial walk. Sides sharing a letter (and color) are the same edge — glue them, matching the arrows, and the polygons reassemble the surface.