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.