UPDATE 2: It turrns out there is an error in this post, concerning the move R3a. The oriented Reidemeister moves which are not respected by the untyped lambda calculus crossings are R2c, R2d, R3a, R3h. All other moves are allowed. [Added: see why by looking at the post Big sets of oriented Reidemeister moves which don’t generate all moves.]
UPDATE: The paper “Graphic lambda calculus and knot diagrams”, arXiv:1211.1604 contains a part of what is discussed here, at the tag “graphic lambda calculus“. There will be follow-ups and applications.
Last post related to the subject is “3D crossings in graphic lambda calculus“. I shall use the notations from there, see also some earlier posts indicated there.
For oriented link diagrams the list of Reidemeister moves is large (24 moves). The list of the moves and a choice of a minimal generating set (with proofs) can be found in the paper by Michael Polyak “Minimal generating sets of Reidemeister moves“.
The following four Reidemeister moves generate all the others.
Let’s use the crossings given by graphic lambda calculus (see the post mentioned at the beginning) and let us interpret the moves as composites of graphic beta moves. Is this possible? Yes, I’ll show you in a moment.
In terms of the graphic lambda calculus, the move R1a is this:
which is just a degenerate graphic beta move with elimination of loops (short name “elimination of loops”).
The move R1b becomes this:
which is another elimination of loops move.
So, the first two moves are just degenerate graphic beta moves with elimination of loops. We may say that we used graphic beta moves, because degenerated moves are much weaker than the full graphic beta move.
Let’s see how R2a looks like in terms of graphic lambda calculus:
That is a combination of two beta moves!
Finally, the move R3a looks like this:
That is a combination of 6 beta moves. The red arrows are those which correspond to the relation over-under of the respective crossings. (see UPDATE 2 for corrections)
Conclusion: the Reidemeister moves, as seen when interpreted in graphic lambda calculus, are combinations of an even number of graphic beta moves and any number of degenerated graphic beta moves with elimination of loops.