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10.3 Amalgamation of cavities

As we have already defined it, a cavity is an empty worm. A cave is an empty dragon.

Under certain circumstances we want to amalgamate two or more cavities into a single cave. This is done before we amalgamate strings. An example where we wish to amalgamate two empty strings is the following:


The two empty worms at a and b are to be amalgamated.

We have already defined a string to be inessential if it meets a criterion designed to guarantee that it has no life potential unless a particular surrounding string of the opposite color can be killed. An inessential string is a string S of genus zero which is not a cutting string or potential cutting string, and which has no edge liberties or second order liberties (the last condition should be relaxed), and which satisfies the following further property: If the string is removed from the board, then the empty worm E which is the worm closure of the set of vertices which it occupied has bordercolor the opposite of the removed string.

Thus in the previous example, after removing the inessential string at the center the worm closure of the center vertex consists of an empty worm of size 3 including a and b. The latter are the boundary components.

The last condition in the definition of inessential worms excludes examples such as this:


Neither of the two X strings should be considered inessential (together they form a live group!) and indeed after removing one of them the resulting space has gray bordercolor, so by this definition these worms are not inessential.

Some strings which should by rights be considered inessential will be missed by this criterion.

The algorithm for amalgamation of empty worms consists of amalgamating the boundary components of any inessential worm. The resulting dragon has bordercolor the opposite of the removed string.

Any dragon consisting of a single cavity has bordercolor equal to that of the cavity.

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