Friday, August 9, 2013

Gems from gems

Action of the map dualities on the edge 4-cycles of a graph-encoded map.

From one graph-encoded map, five other gems are easily generated. The six mutually related gems are called direct derivates.

Recall that black-pink cycles in gems are special. They are always 4-cycles because they represent edges (edges must have exactly two ends) so we cannot make a move that turns black-pink cycles into any other sort of cycle for fear that what will be left (to turn into black-pink cycles) will not have cycles of 4.

We can, however, switch those two colors, black and pink. That will still leave us with a Tait-colored graph having black-pink cycles of length 4. But, in consequence of switching black and pink, the pink-white cycles transform into black-white cycles, and the black-white cycles transform into pink-white cycles. In other words, vertices have become faces, and faces have become vertices. This color switch is an operation of order two, since repeating the color switch just returns the original gem. The new gem encodes what is known as the (Poincare) dual of the original map.

There are four more possible permutations of three colors, but none of them is fair game because of the need to insure that we have black-pink cycles are of length 4. Nonetheless, we can still do some systematic re-connecting of the nest.

Tracing around a black-pink cycle, we can always label the vertices in the order they are encountered such that the colored cycle is:

(ab), {bc}, (cd), {da}

where parentheses are black edges and curly brackets are pink edges. The operation skew "cross-wires" the black segments to make the cycle:

(ac), {cb}, (bd), {da}

The result of skew is called the Petrie dual of the map. 

 The corresponding operation that "cross-wires" the pink segments is called phial, the result of phial is called the antimap. (Since dual allows us to switch colors at will, having a separate, color-specific operator for pink edges is not strictly necessary.)

Like dual, skew and phial are operations of order two: "cross-wiring" the same pair of edges twice leaves them in their original arrangement. Operations of order two are termed dualities.

Composing the three dualities discovered above (Du, Sk, Ph) yields two more operations we can perform on a gem: Sk(G*) and Ph(G*). These two composite operations are mutual inverses—each undoes the other. They are also trialities: applying either operation three times in succession returns the original gem.

Above is an incomplete Cayley diagram (the actions of the two trialities are omitted) that shows what happens to the black-pink 4-cycles under the action of the dualities. (Note the surprising fact that when one color is already crossed, the operation of crossing the other color is equivalent to simply rotating the diagram 90 degrees.)

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