Prelude: First of all, let "S matrix" denote "an abstract 4D tensor satisfying the usual isotypy rules (with no arrows!)". I'm busy trying to classify all possible S matrices (paper pending) - it's just computation and no topology :-)
Lately, I was playing with Kuperbergs G2/B2 constructions. As standalones (defining some reduction rules for trivalent graphs and prove they define an invariant) they might be one of the few possible, but I'm working the reverse way: Again I start with a 3D tensor representing a trivalent node and check the rules by stupid computing.
But I said Birman-Wenzl style: All rules I write using 2+2 tensors (see my "Braid*Temperley-Lieb=?" thread), so we now have S (the braid generator), H (the Temperley-Lieb generator) and U, the spider generator. It looks this way: |=| The double line just signifies it's something different from a single line :-) Technically, U is the tensor product of two trivalent nodes: |=| = >= * =<. (Only the second = is an equal sign :-) Rotate U by 90 deg to get T, now our elements are complete and we can list the basic rules: Pic (Completed by the whirl type rules - like H1H2T1=H1U2 - and the Reidemeister 3 type rules - S1S2U1=U2S1S2 etc.) You have (for now) the free parameters o,z,w,W,a,A,B.
As you might have noted, in this approach there are no natural rules for reducing 4- and 5-gons. But again, since I start with the tensors, I could work out backwards what the rules would be. But the nice thing is: I can take almost (see below!) any working S matrix, and for each of the eigenvalues of S I can find a working U. This gives tons of B2 spider analogues (of course "the" B2 spider is among them).
But here comes my question. One type of S matrix refuses to be augmented
to a spider. I hypothesize all below characterizations are equivalent:
- Let t be a matrix you can "pull a line over": t1S1S2=S1S2t2.
Then t must NOT necessarily be the representation of some actual tangle
(one you can actually cut out of a link).
- Not for every 2+2 tangle t (cut out of a link) S*t=t*S.
- Or: The S matrix can distinguish mutants.
- More algebraically: The infinity tangle H can not be written as a sum of
finite many powers of S.
Does any of the characterizations ring a bell? Here is an example S: