# Minimal set of geometric moves in various equivalence classes of triangulated geometries

I would like to get to know what is the minimal set of geometric changes "aka. moves" (topology preserving modifications / Pachner moves / bistellar moves) that can transform any 3-dimensional triangulations (with given constraints) into any other.

Under given constraints, I mean different equivalence classes of geometries.

The best way to discuss it is via the dual graph of the triangulation, where every node is a tetrahedron and every edge is a face.

A: combinatorial triangulation: Every node is strictly 4-valent, and one can have only one dual-edge between nodes.

B: relaxed (nondegenerate): there can be multiple connections between nodes, but every face and similarly every edge is defined by 3 different vertices. Similarly, a triangle is defined by 3 different edges and a tetrahedron is by 4 different triangles. There are no self-connections allowed in the dual graph.

C: degenerate: allows for self-adjacency, nodes can connect back to themselves in the dual graph.

Taking the simplest case, the triangulation of S^3 from now on.

I assume, that a {1,4} and a {2,3} move can be enough to reach any geometric configurations in the case of A. Move {1,4} replaces a node with 4 half-dual-edges with 4 nodes that are fully connected except 1-1 remaining half-dual-edges. Move {2,3} exchanges two nodes that share a dual-edge (and have 3-3 half-dual-edges) with 3 fully connected nodes with 2-2 half-dual-edges each.

Case B is a bit more complicated, instead of move {1,4} let's use a {0,2} move, where a dual-edge between two nodes is replaced by 2 fully connected nodes (3 connections) plus 1-1 half-dual-edges. I thought, that these two moves should function, however, I couldn't yet figure out a simple situation:

Let's define a new move: take a loop in case B, cut it in half (introducing 4 half-dual-edges), place down 2 nodes with 2 connected dual-edges (thus 2-2 external ones), then connect one half of the loop to one and the other to the other node. This is a {0,2} move, but it is geometrically more involving.

In the simplest situation, imagine a graph of 2 maximally connected nodes (call it initial configuration). Performing this move turns this into a graph, which has 4 nodes, forming a circle with 2-2 dual-edges between each "adjacent" node.

How can one reach this configuration without using the newly defined {0,2} geometric move?

Situation C is even more complicated, thus I would restrict my question to situation B only.

• What about Pachner moves? Pachner, Udo (1991), "P.L. homeomorphic manifolds are equivalent by elementary shellings", European Journal of Combinatorics, 12 (2): 129–145, doi:10.1016/s0195-6698(13)80080-7 Aug 15, 2023 at 17:39
• Unfortunately I don't understand the paper (entirely). I'm not too familiar with abstract math, and it's hard for me to read it. I cannot find answer to my question in this paper. Aug 15, 2023 at 19:42
• @Kregnach: You should try sending your question via email to Jaco and Rubinstein. They likely have thoughts on most of your questions. Aug 16, 2023 at 6:35
• Finally, this paper (arxiv.org/abs/1812.02806) gives work in the direction you desire. If it does not answer your question, then you might reach out to the mathematicians mentioned in the abstract. Aug 16, 2023 at 10:24
• @SamNead the paper you suggested is really great and useful, thank you! Aug 16, 2023 at 11:49