Zipper logic and knot diagrams

In this post I want to show how to do zipper logic with knot diagrams. Otherwise said, I want to define zippers and their moves in the realm of knot diagrams.

Knot diagrams, it’s a way of saying, in fact I shall use oriented tangle diagrams (i.e. the wires are oriented and there might be in or out wires) which moreover are only locally planar (i.e. we admit also “virtual crossings”, in the sense that the wires may cross without creating a crossing 4-valent node) and not, as usual, globally planar.

Here is the half zippers definition:

zipper_loop_1

As you see, each half zipper has some arrows which are not numbered, that is because we don’t need this information, which can be deduced from the given numbering, just by following the arrows.

We have to define now the CLICK move. For the zippers from chemlambda, that was easy, the move CLICK is trivial there. No longer here:

zipper_loop_2

The figure illustrates a CLICK move between a (-m)Z and a (+n)Z with m>n.  It’s clear how to define CLICK for the other cases m = n and m<n.

Finally, the ZIP move is nothing by repeated application of a R2 move:

zipper_loop_3

It works very nice and it has a number of very interesting consequences, which will be presented in future posts.

For the moment, let me close by recalling the post Two halves of beta, two halves of chora.

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