It looks that there is a connection of the following with the research on DNA topology, but I shall comment on this in a future post, mainly because I am learning about this right now. (But another main reference is Louis Kauffman.)
If I am using this encoding of chemlambda nodes with crossings
which is a variant of the encoding from Curious crossings then, as in the mentioned post, the beta move becomes a CLICK between “sticky ends” which are marked with dashed lines, followed by R2b move, and also the FAN-IN move becomes a CLICK between sticky ends, followed by a R2a move.
What about the DIST moves? Let’s take the DIST move which involves an application node and a fanout node. The move looks like this:
In the right column we see the DIST move with chemlambda drawing conventions. In the left column there is the translation with crossings and sticky ends.
What do we see? The strings 1-5 and 6-3 are transposed by the DIST move and a new string appears, which crosses them.
I can draw the same move like this:
This is starting to look interestingly alike some other pictures from DNA topology.