Related: (Graphic) beta rule as braiding.
Louis Kauffman proposes in his book Knots and Physics (part II, section “Slide equivalence”), the notion of slide equivalence. In his paper “Knotlogic” he uses slide equivalence (in section 4) in relation to the self-replication phenomenon in lambda calculus. In the same paper he is proposing to use knot diagrams as a notation for the elements and operation of a combinatory algebra (equivalent with untyped lambda calculus).
There is though, as far as I understand, no fully rigorous relation between knot diagrams, with or without slide equivalence, and untyped lambda calculus.
Further I shall reproduce the laws of slide equivalence (for oriented diagrams), following Kauffman’ version from Knots and physics. Later, I shall discuss about the relations with my graphic lambda calculus.
Here are the laws, with the understanding that:
1. the unoriented lines may have any orientation,
2. For any version of orientation which is depicted, one may globally change, in a coherent way, the orientation in order to obtain a valid law.
Law (I’) is this one:
Law (II’) is this:
Obviously, we have four gates, like in the lambda calculus sector of the graphic lambda calculus. Is this a coincidence?
UPDATE (06.06.2014): The article Zipper logic arXiv:1405.6095 answers somehow to this, see the post Halfcross way to pattern recognition (in zipperlogic).A figure from there is:
See also the posts Curious crossings (I) and Distributivity move as a transposition (Curious crossings II).