Problems with slide equivalence

UPDATE: System X is a solution.

_________

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.