Recall the fact that the representations of a quantum group form a braided tensor category, and this corresponds to the fact that $U_q(\mathfrak g)$ is a quasi-triangular Hopf algebra. The braiding operations can be constructed via the KZ equations for representations of $\mathfrak g$. The first thing that the KZ equations give is a quasi-triangular quasi-Hopf algebra, i.e. we have an associator present. It turns out that this is equivalent to $U_q(\mathfrak g)$ in some suitable sense, but one needs to somehow absorb the associator into the rest of the structure in order to get a genuine quasi-triangular Hopf algebra (and this is not straightforward to do).

The Kontsevich Integral for tangles can be calculated in same the formal way one calculates the monodromy of the KZ equations (of course, since this is just the definition of the Kontsevich integral). Thus to calculate the value of the Kontsevich Integral for some parenthesized tangle, it suffices to multiply the values of together the invariant for elementary tangles and changes of parenthization (this being the use of the associator) corresponding to the given input tangle. These can be given more or less explicitly, as the (chord diagram)-valued "R-matrix" and Drinfeld associators respectively.

Is there a (chord diagram)-valued R-matrix (hopefully somewhat explicit) which does not require an associator? (i.e. one that genuinely satisfies the relations satisfied by the elementary tangles in the category of unparenthesized tangles)?

One would certainly need to redefine the values of the invariant of min/max elementary tangles as well.

  • $\begingroup$ This seems highly unlikely to me. My approach to rule it out would be to look at simple knot and link diagrams to constrain what the chord-diagram valued R-matrix could possibly be. $\endgroup$ – Jim Conant Oct 18 '11 at 22:16
  • $\begingroup$ You might also be interested in Dancso's paper, which avoids the use of associators in the construction of a universal invariant (it's written for KTG's, but they replace q-tangles, and seem like a better formalism), and is quite elegant: arxiv.org/abs/0811.4615 $\endgroup$ – Daniel Moskovich Mar 12 '12 at 14:40
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    $\begingroup$ I know this is nitpicky, however: $U_q(\mathfrak{g})$ is not a quasitriangular Hopf algebra in the true sense since the R-matrix does not live in the algebraic tensor product $U_q(\mathfrak{g}) \otimes U_q(\mathfrak{g})$. Of course the R-matrix acts in tensor products of finite-dimensional representations, and this gives the braiding, etc. But I think it pays to be careful about how one refers these things. $\endgroup$ – MTS Jun 24 '12 at 18:10

As you point out, the relation between associators and the quasi-triangular structure of $U_q(\mathfrak g)$ (and the related tangles invariants) exists "only" at the Lie algebraic level, not (not yet) at the universal one. Roughly speaking, this is because the twisting which absorb the associator is not $\mathfrak g$-invariant, ie modifies the coalgebra structure, which a priori doesn't make sense at the level of chord diagrams.

So far I know, there is no combinatorial construction of a universal finite type invariant which can avoid associators. But of course it's something people are looking for.

But it turns out that the theory of quantum $R$-matrices is more related to the theory of virtual knotted objects (see this answer of Greg Kuperberg). This is more or less "Bar Natan's dream" that a universal finite type invariant for virtual knotted objects should corresponds somehow to Etingof--Kazhdan quantization of Lie bialgebras.

There is also a baby version of this, which is Alekseev-Enriquez-Torrossian solution of the Kashiwara Vergne conjecture based on associators. It turns out that they constructs a kind of universal twist which can "kill" the associator, and a kind of universal solution of the quantum Yang-Baxter equation, living in a bigger algebra than the algebra of horizontal chords diagrams. According again to Bar Natan, this corresponds more or less to a universal finite type invariant for "welded knots". See: http://www.math.toronto.edu/drorbn/papers/WKO/

You may also find this paper interesting : Towards a Diagrammatic Analogue of the Reshetikhin-Turaev Link Invariants

Edit: Some details about the relation between the Alekseev-(Enriquez)-Torossian construction and Vassiliev invariants.

They start from the Lie algebra $\mathfrak{tder}_n$ of "tangential derivations" of the free Lie algebra $\mathfrak{f}_n$ on $n$ generators, that is the Lie algebra of endomorphism sending each generator $x_i$ to $[x_i,a_i]$ for some $a_i \in \mathfrak f_n$.

Let $r^{i,j}$ be the element mapping $x_i$ to $[x_i,x_j]$, and $x_k$ to 0 for $k\neq i$. Then it leads to a solution of the "classical Yang-Baxter equation whose second leg lives in a commutative subalgebra", i.e.: $$[r^{1,3},r^{2,3}]=0$$ and $$[r^{1,2},r^{1,3}+r^{2,3}]=0$$ as does, for example, the $r$-matrice of the Drinfeld double of a cocommutative Lie bialgebra. They also prove that $R=\exp(r)$ ($r:=r^{1,2}$) satisfies the Quantum Yang Baxter equation. Therefore, one get this way a representation of the (pure) braid group in $\exp(\mathfrak{tder}_n)\rtimes S_n$. The $r^{i,j}$ can be thought of as "arrow diagrams" and are related to the welded braid group.

On the other hand, exactly like in the usual theory of the Yang-Baxter equation, we have that $t^{i,j}=r^{i,j}+r^{j,i}$ satisfies the infinitesimal braid relations, also called 4t relation for horizontal chord diagrams. We thus get an injective morphism from the algebra of Horizontal chord diagram into $U(\mathfrak{tder}_n)$. Therefore we can put an associator and get another representation of the braid group in $\exp(\mathfrak{tder}_n)\rtimes S_n$ which is precisely the "Kontsevich integral for braids".

Then the main result of [AT] is the following identity: let $\Phi$ be (the image in $\exp(\mathfrak{tder}_3)$ of) an associator, then there exists some $F \in \exp(\mathfrak{tder}_2)$ such that $$F^{2,3}F^{1,23}=\Phi F^{1,2} F^{12,3}$$

where the indices correspond to some maps $\mathfrak{tder}_2 \rightarrow \mathfrak{tder}_3$ modelled on the coproduct of an envelopping algebra. They also show that:

$$R=F e^{t/2} (F^{2,1})^{-1}$$

Hence $F$ is a universal version of a Drinfeld twist for Quasi-Hopf algebra, which is able to "kill" the associator, and according to the theory it implies that the corresponding representations of the braid group are the same.

Edit 2: This was actually the situation for braids, let me add a few word about knots. It turns out that while usual braids embeds into welded one, this is far from being true at the level of knots. So the twist also intertwines between the restriction of the invariant for welded knots to usual knots, and the image of the Kontsevich integral in the space of arrow diagrams, but the resulting invariant is much weaker.

Welded braids can be identified with the group of basis conjugating automorphisms of a free group, and the map it gets from usual braids is nothing but the Artin representation. This representation is faithfull, but its extension to string links (and in particular to long knots) has a huge kernel. At the level of diagrams, the algebra corresponding to knots is free commutative with two generators in degree one, and one generator in each degree greater than one. Hence it has a rather simple structure, while its analog for usual knots is very complicated.

In fact, one of the main claim of Bar Natan's paper is that the universal invariant for welded knots is roughly the Alexander polynomial.

However, let me mention that usual knots does embed into virtual one, and that the above story is an important step towards something similar in the virtual case.


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