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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.

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  • $\begingroup$ 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 $\endgroup$ Aug 15, 2023 at 17:39
  • $\begingroup$ 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. $\endgroup$
    – Kregnach
    Aug 15, 2023 at 19:42
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    $\begingroup$ @Kregnach: You should try sending your question via email to Jaco and Rubinstein. They likely have thoughts on most of your questions. $\endgroup$ Aug 16, 2023 at 6:35
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    $\begingroup$ 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. $\endgroup$
    – Sam Nead
    Aug 16, 2023 at 10:24
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    $\begingroup$ @SamNead the paper you suggested is really great and useful, thank you! $\endgroup$
    – Kregnach
    Aug 16, 2023 at 11:49

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