Vertex split-merge operation
By
multiplying permutation by transposition from left side in the permutation two
orbits either merge into one or one becomes divided into two orbits depending
on whether elements of transposition are in two distinct orbits or both in one
orbit:
in the
first case, when orbits merge or:
in the
second case, when orbit is split into two new orbits,
or, using
conjugate operator :
.
Applying
this operation to combinatorial map, its vertices are either merged or split.
In the picture it is illustrated how it looks like geometrically. Let and .
The result we get is equal to . Let us take a note of the fact that in the formula indices
would arrange more symmetrically if we chose to number corners not clockwise,
but anticlockwise [with considering as if
standing before other corners].
This geometrical
interpretation of multiplication of a single transposition gives an interesting
graph-theoretical result.
Let us call
the operation corresponding to corner split-merge operation
Theorem: Producing
corner split-merge operation with all pairs of corners from edge rotation we
get the dual graph of the graph.
Proof of
the fact is trivial from combinatorial point of view. But, graph-theoretically
it gives an impression of some magic. Let us look in an example what goes on.
Tetrahedron
with face rotation and edge rotation .
Let us
apply to tetrahedron the operation of multiplying from the left transpositions
from edge rotation.
Let us
multiply by first transposition from edge rotation: . We get:
Let us multiply by second transposition from
edge rotation: . We get:
Let us
multiply by third transposition from edge rotation: . We get:
Let us
multiply by fourth transposition from edge rotation: . We get:
Let us
multiply by fifth transposition from edge rotation: . We get:
Let us
multiply by sixth transposition from edge rotation: . We get:
This map is
dual map to previous map, i.e., its vertex rotation is equal to face rotation
of the previous map.