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I am sorry, but I don't understand the question. The connectivity layers are stored inside the technology setup. You can export or import technology setups, if that is what you mean. Please see http://klayout.de/doc/about/technology_manager.html for details.
The current link to the documentation is here:
I was also looking for the file in which the "Layer Stack" information is stored and finally found it in my local user's klayoutrc file which I found at