# Small graph rewrite systems (4)

This post follows Problems with slide equivalence. A solution is to replace slide equivalence with System X.

This supposes to change the decomposition of a crossing like this:

I let you discover system X (or will update later) but here I want to show you that the Reidemeister 3 rewrite looks like that:

There is now a page dedicated to small graph rewrite systems and stick-and-rings graphs.

# Problems with slide equivalence

UPDATE: System X is a solution.

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After the Intermezzo, in this post I’ll concentrate on the slide equivalence for unoriented (virtual) links, as defined in L.H. Kauffman, Knots and Physics, World Scientific 1991, p. 336.

Later on we shall propose a small graph rewrite system which is different from this, but we first need to understand that there are some problems with slide equivalence.

Kauffman rule I’ is half a definition, half a rewrite rule. He gives two decompositions of a crossing into two 3-valent nodes. The rewrite is that we can pass from one decomposition to the other.

Problem 1. How many types of 3-valent nodes are used? My guess is just one.

Problem 2. Is the rule II’ needed at all? Why not use instead the rule III’, with the price of a loop:

Problem 3. Is the rule I’ too strong? Maybe, look at the following configuration made of two crossings.

Neighboring crossings dissappear.

We don’t even need two neighboring crossings. In the next figure I took the left pattern from the rule IV’, first part. It is also a pattern where the rules I’, then III’ apply.

The result is very different from the application of IV’.

The same happens for the right pattern of the rule IV’, first part.

We can use again I’ and III’ to obtain a very different configuration than expected.

Conclusion.  The slide equivalence rewrites with a “dumb” algorithm of rewrites application behaves otherwise than expected. By “dumb” I mean my favorite algorithms, like greedy deterministic or random.

Used with intelligence, the slide equivalence rewrites have interesting computational aspects, but what about the “intelligent” algorithm? Kauffman brains are rare.

# 3D crossings in graphic lambda calculus

Related:

Let us look again at the NOTATIONS I made in the post (A) for crossings in graphic lambda calculus:

When seen in 3D, both are the same. Indeed, the 3D picture is the following one:

How to imagine the graphic beta move? Start with two wire segments in 3D, marked like this:

Glue the small blue arrow (which is just a mark on the wire) which goes downwards away from the blue wire with the small red arrow which goes downwards to the red wire:

That’s the graphic beta move, in one direction. For the opposite direction just rewind the film.

There is a slight  resemblance with the figures from the post (B), concerning slide equivalence, consisting in the fact that here and there we see crossings decomposed (or assembled from) two types of gates, namely one with one entry, two exits, the other with two entries, one exit.

Notice also that in graphic lambda calculus we have another two gates, namely the FAN-OUT gate and the TOP gate. We shall see how they couple together, next time.

# Slide equivalence of knots and lambda calculus (I)

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:

Law (III’):

Law (IV’):

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).

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