Suppose I have a 3-manifold $M$, and I write down a triangulation $T$ of $M$. Then I can also create a graph $G$ dual to this triangulation, with a node of the graph for every tetrahedron of $T$, and an edge between two nodes in $G$ for every triangular face shared by two tetrahedra in $T$. If I require that the connections of the edges in $G$ have fixed locations on the nodes, then I can define moves acting on the nodes and edges of $G$ which are equivalent to the Pachner moves on $T$. From this, I can find new graphs $G'$ which are each dual to a new triangulation $T'$ of a manifold homeomorphic to $M$.
My questions relate to the extent you can go in the other direction -- from a 4-valent graph to a triangulation of a manifold. Specifically, suppose I have a graph with all nodes having four edges embedded in $\mathbb{R}^3$, so I can have at least some of the edges be part of non-trivial braids.
- Are there any such braided graphs that correspond to the triangulation of a 3-manifold?
- If so, what are the restrictions on the braids that allow the duality to work both ways?
- This question is vaguer, but I am also curious about what exactly "fails" for the braided graphs that do not correspond to a 3-manifold triangulation? In other words, what obstruction due to the braid is preventing the formation of the dual triangulation from a braided 4-valent graph?
Any input or references to these equations would be greatly appreciated.
This is not an answer as such, but just some ideas on where to go with this. Along with my co-authors, I have looked at a very similar problem. We took the dual graph and determined exactly how to "flesh it out" so that it fully encoded the information on a particular triangulation. That is, there is a 1-to-1 correspondence between triangulations and the graph setting that we devised.
Just with the wording, I tend to use arc to mean 'edge of the dual graph' and node to mean 'vertex of the dual graph' to make things a bit clearer when reading.
In the new setting, we replace each arc of the dual graph with a "triple arc" such that each individual arc represents two edges (of distinct tetrahedron faces) which are identified in the triangulation. Then various constraints (such as the 3-manifold constraints) can be converted into this new setting.
The paper is up on arXiv at http://arxiv.org/abs/1412.2169 if you want to read further.