Exchange and expansion: Dual versions of the Pachner moves

Expansion and exchange moves, dual versions of the Pachner moves. Image quoted from Bilson-Thompson et al., "Update on braids and preons."

Each of the Parchner moves (and their compositions) has an equivalent in the dual map, the 3-regular (a.k.a., trivalent or cubic) map that directs the connectivity of the triangles in the triangulation.

Exchange = dual version of flip₂₂ = Find a 3-edge path that turns left-then-right or right-then-left. Slide the two non-path, branch edges past each other. Leaves the face, vertex, and edge counts unchanged. In fullerene chemistry this is the Stone-Wales transformation.

Expansion = dual version of flip₁₃ : truncate a (trivalent) vertex, creating a trianglular face in its place. Increases face count by 1, vertices by 2, and edges by 3.

Contraction = dual version of flip₃₁ = contract a trianglular face, leaving a trivalent vertex in its place. Decreases face count by 1, vertices by 2, and edges by 3.