Dual of the graphic beta move implies some Reidemeister moves

For the extended beta move see the dedicated tag.  See also the tutorial “Introduction to graphic lambda calculus“.

The extended beta move is still in beta version. In this post I want to explore some consequences of the dual of the graphic beta move.

I proved earlier the the extended beta move is equivalent with the pair (graphic beta move, dual of the graphic beta move), let me recall this pair in the following picture.

beta_beta_star

Here are some particular applications of the dual of the graphic beta move. The first is this:

beta_beta_star_1

But this is equivalent with the emergent algebra move R1a (via a CO-COMM move).  Likewise, another application of of the dual of the graphic beta move is this:

beta_beta_star_2

which is the same as   the emergent algebra move R2a.

The third application is this one:

 

beta_beta_star_3

 

and it was discussed in the post “Second thoughts on the dual of the extended beta move“, go there to see if this move may be interpreted as a pruning move (or an elimination of loops).

Finally, there is also this:

beta_beta_star_4

which was not mentioned before. It suggests that

beta_beta_star_5

behaves like the generic point in algebraic geometry.

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