Curious crossings

I’m coming back to  the post   Halfcross way to pattern recognition (in zipperlogic)    and I am going to modify the drawings a bit.

Instead of this convention of transforming chemlambda nodes into half-crossings

halfcross_1

I draw this mapping from chemlambda nodes to crossings:

cdeco_1

Remark that at the left we have now crossings. At the right we have the chemlambda nodes, each with a dashed half-arrow attached, so now the nodes become 4-valent locally planar ones.

As usual (here at chorasimilarity) the crossing diagrams are only locally planar!

Look closer: there are two kinds of oriented crossings, right? To each kind corresponds a colour (green or red) of a chemlambda node.

This is no longer a dictionary, there is no longer a bijective correspondence, because for each oriented crossing there are two possible chemlambda nodes, depending on where is the dashed half-arrow! That is the meaning of the wiggling blue arrow from right to left, it’s uni-directional.

OK then, instead of drawing this interpretation of the beta move

halfcross_2

we get the following one

cdeco_2

where the arrows are drawn like that in order to see what transforms into what (otherwise there is no need for those convoluted arrows in the formalism).

Likewise, instead of this

halfcross_3

I draw this:

cdeco_3

Funny, what could that mean?

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2 thoughts on “Curious crossings”

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