Let us start in 3D. Say I give you a polytope cone and you slice one of its edges out, like in this picture:
(i.e. you grab a plane that passes through it's apex and cut off a single edge)
Let's make the problem even simpler. Say the cone is a simple cone, i.e. 3 faces, 3 edges and label the faces $f_1, f_2, f_3$.
The edges can be defined as the intersections of 2 faces $e_1 = f_1 \cap f_2$ and analogously for $e_2$. For shorthand let's say $e_{ij} = f_i \cap f_j$.
If the plane cuts off the edge $e_{23}$ then it means it intersects the cone into 2 edges, one by intersecting $f_2$ and one by intersecting $f_3$.
Now let's say we are in 4D.
An edge in 4D will be the intersection of 3 hyper planes. So let us say that $e_234$ is the edge that we are cutting from the simple cone. Let $p$ denote the new plane. The question is if you can generate a sequence of symbols containing $p$ so that every contiguous sequence of 3 elements in it is an edge of the poltyope cone (non encessarily exhaustive).
So for example, going back to 3D. The original cone was $123$ and after cutting it it became $12p3$ because $12$ is a valid edge, so are $2p, p3, 31$.
In 4D our plane generates 3 new edges, we know that it intersects the planes $2,3,4$. We are missing which other planes generate the newly introduced edges.
Am I correct to say that $23p45$ is the sequence of symbols that corresponds to the 3 new edges? such that $23p, 3p4, p45$ are the 3 edges in 4D?