I arrived to Penrose's paper Applications of negative dimensional Tensors after reading some bits of Baez's Prehistory (link) and the first two chapters of Turaev's Quantum invariants of knots and 3-manifolds (link). The main result in the last one is a presentation of $\text{Rib}$ (the category of ribbon graphs) by a set of generators and relations (braiding, twist and their inverses, duality morphisms...).

On the other hand Penrose seems to add to the natural braided ribbon structure also a differential one finding generators also for the (anti)symmetrization and covariant-derivative operations. Are there some easily-found references for a structure theorem similar to Lemma 3.1.1 in the book of Turaev?

Edit: I'm sorry I forgot the second question: I have some problems in figuring out the "contraction" in Penrose notation, because in Turaev exposition the juxtaposition of "coupons" seems to be reserved to the composition of morphisms (contraction contrarily implicates some sort of Einstein summation, am I right?)

This is the last edit, I promise: the following convention to contract the sum of two tensors

enter image description here

doesn't make any sense to me.. How can the "f" leg of $\chi^b_{fde}$ split into two?

  • 1
    $\begingroup$ In your bottom picture, there is one f leg for the bracketed sum which arises from an f leg from $\theta$ and one from $\gamma$. That's where the apparent "splitting" comes from. $\endgroup$
    – j.c.
    Sep 23 '11 at 17:45
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    $\begingroup$ I have not looked at these diagrams in years, but the $f$ leg of $\chi^b_{fde}$ is split in two due to the $+$ between $\theta^{af}_c$ and $\gamma^{af}_c$. The lower $f$ index of $\chi^b_{fde}$, the triangle, is being contracted with the upper $f$ index of $\gamma^{af}_c$, the square, and the result is being summed with the result of contracting the lower $f$ index of $\chi^b_{fde}$, the triangle, with the upper $f$ index of $\theta^{af}_c$, the circle. This split corresponds to the sum $\theta^{af}_c + \gamma^{af}_c$. If there were another summand, the $f$ leg would split in to three pieces. $\endgroup$ Sep 23 '11 at 17:51
  • $\begingroup$ cxontraction is a bilinear operation, isn't it? But try to write down the sum $\theta_c^{af}\chi_{fde}^b+\gamma_c^{af}\chi_{fde}^b$; it is not at all immediate that this diagram is (isotopy?) equivalent to that I linked in my question... $\endgroup$
    – fosco
    Sep 23 '11 at 17:58
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    $\begingroup$ the juxtaposition of "coupons" seems to be reserved to the composition of morphisms A covariant (or mixed) tensor is a linear transformation, which is a morphism. For example, $M^a_b N^b_c$ can be interpreted as a notation for matrix multiplication, which is the composition of two morphisms. contraction contrarily implicates some sort of Einstein summation, am I right? Penrose's birdtracks notation is isomorphic to abstract index notation. Indices don't take integer values or imply a choice of basis. Summation only happens if you convert to concrete index notation. $\endgroup$
    – user21349
    Aug 18 '15 at 18:42

To the best of my knowledge, no abstract formulation and soundness theorem of the full Penrose notation (with symmetriser, antisymmetriser, covariant derivative, etc) exists. There is, however, a veritable zoo of graphical languages for various kinds of monoidal categories (with a regrettably small subset of them having published soundess/completeness proofs). The best place to start is Selinger's survey paper from 2011:



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