Big sets of oriented Reidemeister moves which don’t generate all moves

The article  “Minimal generating sets of Reidemeister moves“ by Michael Polyak gives some minimal generating sets of all oriented Reidemeister moves. The question I am asking in this post is this: how big can be non-generating sets of moves?

The oriented Reidemeister moves are the following (I shall use the same names as Polyak, but with the letter \Omega  replaced by the letter R):

  • four oriented Reidemeister moves of type 1:

reidemeister_1

  • four oriented Reidemeister moves of type 2:

reidemeister_2

  • eight oriented Reidemeister moves of type 3:

reidemeister_3

Among these moves, some are more powerful than others, as witnessed by the following

Theorem. The set of twelve Reidemeister moves formed by:

  • four moves of type 1
  • two moves of type 2, namely R2a and R2b
  • six moves of type 3, namely R3b, R3c, R3d, R3e, R3f, R3g

does not generate the remaining moves R2c, R2d, R3a and R3h.

______________

I postpone the proof for another time, here I just want to emphasize that the moves R2c, R2d, R3a and R3h are more special than the rest of them. The proof which I have is via the graphic lambda calculus, but probably other proofs exists. Please tell me if you know a proof of this theorem.

UPDATE: see a proof at mathoverflow.

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