3D crossings in graphic lambda calculus

Related:

(A)  (Graphic) beta rule as braiding

(B)  Slide equivalence of knots and lambda calculus (I)

(C) Braitenberg vehicles, enchanted looms and winnowing-fans

Let us look again at the NOTATIONS I made in the post (A) for crossings in graphic lambda calculus:

When seen in 3D, both are the same. Indeed, the 3D picture is the following one:

How to imagine the graphic beta move? Start with two wire segments in 3D, marked like this:

Glue the small blue arrow (which is just a mark on the wire) which goes downwards away from the blue wire with the small red arrow which goes downwards to the red wire:

That’s the graphic beta move, in one direction. For the opposite direction just rewind the film.

There is a slight  resemblance with the figures from the post (B), concerning slide equivalence, consisting in the fact that here and there we see crossings decomposed (or assembled from) two types of gates, namely one with one entry, two exits, the other with two entries, one exit.

Notice also that in graphic lambda calculus we have another two gates, namely the FAN-OUT gate and the TOP gate. We shall see how they couple together, next time.

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