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 replaced by the letter ):
- four oriented Reidemeister moves of type 1:
- four oriented Reidemeister moves of type 2:
- eight oriented Reidemeister moves of type 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.