Consider the algebra A on algebraic tangles t (t described on the Wiki page -
Wiki Tangles ) with the
usual operations reflection, addition, product, ramification.
Make the additional rule that t=Mutant(t). The so-defined A nearly looks like
a Boolean algebra and you just need e.g. product and reflection.
With a trick you can write each algebraic tangle even as a Catalan binary string:
11=0,t0=~t, so you don't need reflection either, and you can even "forget" the
closing bracket and the last 1 in the notation of a tangle: 0=(11) =: (1 ,
-1=(10)=(1(11)) =: (1(1, inf=(01)=((11)1) =: ((11 and so on.
Now my question: There are a lot of identities in this algebra. For example
0t=inf for all t. But is it possible to run a pattern detector over the
string which reads e.g. 00 =: ((11(1 and replaces it with ((11, and shortens every
string to minimum length? Obviously 1. there would have to exist a finite
rule system R, 2. R may not run into false minima, including equivalent
equal-length words. Local minima exist for Reidemeister moves, as you know.
If R exists, one could derive from it an ultra-fast knot polynome computer for algebraic links. (Many knot polynomes don't distinguish mutants anyway.)
Maybe the Yu paper in Math.Ann. already deals with this, but SpringerLink's PDF crashes at current time. (And I wouldn't understand the paper anyway, I fear :-) Yu paper
Hauke
