Tag Archives: system X

Small graph rewrite systems (5)

Here are some more tentative descriptions of system X and a play with the trefoil knot. This post comes after the intermezzo and continues the series on small graph rewrite systems.

Recall that system X is a proposal to decompose a crossing into two trivalent nodes, which transforms a knot diagram into an unoriented stick-and-ring graph.

2cols-spin-x

The rewrites are the following, written both with the conventions from the stick-and-ring graphs and also with the more usual conventions which resemble the slide equivalence or spin braids mentioned at the intermezzo.

The first rewrite is GL (glue), which is a Reidemeister 1 rewrite in only one direction.

2cols-spin-gl-x

The second rewrite is RD2, which is a Reidemeister 2 rewrite in one direction.

2cols-spin-rd2-x

There is a DIST rewrite, the kind you encounter in interaction combinators or in chemlambda.

2cols-spin-dist-x

And finally there are two SH rewrites, the patterns as in chemlambda or appearing in the semantics of interaction combinators.

2cols-spin-sh1-x

2cols-spin-sh2-x

One Reidemeister 3 rewrite appears from these ones, as explaned in the following figure (taken from the system X page).

2cols-spin-rd3

Let’s play with the trefoil knot now. The conversion to stick-and rings

2cols-spin-3foil

is practically the Gauss code. But when we apply some sequences of rewrites

2cols-spin-3f-2

we obtain more complex graphs, where

  • either we can reverse some pairs of half-crossings into crossings, thus we obtain knotted Gauss codes (?!)
  • or we remark that we get fast out of the Gauss codes graphs…

thus we get sort of a recursive Gauss codes.

Finally, remark that any knot diagram has a ring into it. Recall that lambda terms translated to chemlambda don’t have rings.

System X, semantic pain and disturbing news to some

This is a temporary post. Soon some news will come, some disturbing for some. This is just to entertain you with the System X, a small graph rewrite system proposed as a replacement for slide equivalence. Here is some prose I wrote while trying to understand 3 tiny graphic beta rewrites. This qualifies as semantic pain, but it was a very good exercice because it gives ideas (to those prone to have them, as opposed to those who lack personal ideas and take them without acknowledgement).

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.

slideq

 

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.

q1

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

q2

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

q3

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.

q4

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

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

q5

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.