Tag Archives: trefoil

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.