Hi,

I hope filing this under "Morse theory" is correct.

For the "abstract tensor" approach to knot theory you need: a) a 4D tensor to emulate particle interaction, b) a 2D tensor to emulate pair creation/annihilation. (I formulate it in Feynman diagram terms.) You don't actually need c) a particle propagator since this is the delta function.

Now you can rotate a knot by 90 degrees and you can rotate
a Feynman diagram by 90 degrees, but you can't "rotate
Morse theory by 90 degrees"! And I find this extremely
annoying. A "nice" abstract tensor Sabcd emulating a
crossing should have *exactly* the same symmetry properties
as a crossing, i.e. invariance under all 180 degree
rotations. And for the pair tensor Pab likewise Pab=Pba (1),
and the delta function should be Pab too, instead.

One *can* introduce "gauges" and use an equivalent tensor
formulation, using Pab for left and right turns,
but (1) gets you into problems in no time and I never was
able to find a "Lorentz-invariant" tensor set.
The interesting solutions are even CPT-invariant (for intuitive
interpretations of C, P, and T for abstract tensors, I even
have charge and baryon number conservation automagically),
but T(S)=S^-1 should hold (where I only can get T(S)=S for my
solutions).

Am I doing something wrong or is Morse theory (or more precise, Kauffman's abstract tensor approach) symmetry-breaking by nature?

Hauke

isn'tinvariant --- Morse theory is a calculational tool, and requires a choice of height function and understanding of how different choices are related, just like the Reidemeister moves are. When Hauke says "Lorentz invariance" he's talking about the SO(2) that acts on knot diagrams. – Theo Johnson-Freyd Feb 24 '11 at 17:56decompositionas such which is the problem: OK, if you rotate a knot by 90 degrees, the pieces will be different. But if I decompose nonstandardly, i.e. assigning a tensor to ALL turns < > ^ v and crossings °/. .\°, instead of letting < and > be the delta function (forgive my ASCII), this is gauge-equivalent to the standard approach. (If < = Pab, > = Qab, °/. = Rabcd, then the standard crossing tensor is sum(Pai*Ribcj*Qjd).) So < > swaps with ^ v and °/. with .\°, andthe tensor itselfshould reflect that symmetry. (cont) – Hauke Reddmann Feb 25 '11 at 15:27