There is a theorem (I believe by Ocneanu) that the Markov trace on the tower of Temperley-Lieb algebras is (essentially) unique.

What about just traces on separate algebras? That is, take one of them, say $\mathbf{TL}_n$; what is the dimension of the space of linear functionals $\operatorname{tr}$ on it with the property $\operatorname{tr}(xy)=\operatorname{tr}(yx)$? (If one considers their versions with free parameters, i. e. as algebras over polynomials in parameters as free variables, then linearity is understood over these polynomials).

Equivalently, this is the question about the rank of the quotient of, say, $\mathbf{TL}_n$ by the subspace (resp. submodule over polynomials) spanned by all elements of the form $xy-yx$. If you want, about the rank of the 0th Hochschild homology (with itself as coefficients).

I could in fact restrict myself to the mother of them all - the group algebra of the braid group on $n$ strings. There, as Qiaochu Yuan says in a comment to the answer, there are as many independent traces as conjugacy classes in the group.

I've played with it a bit in lower dimensions, and got impression that the trace might be still essentially unique.

However as the answer below shows, this is certainly not so already for $\mathbf{TL}_4$ over $\mathbb C$.

Just for fun, here is the corresponding table of the (nontrivial cases of) "$xy=yx$" for $\mathbf{TL}_3$.

STILL LATER

Looked more carefully at the case of $\mathbf{TL}_4$; one sees that there are indeed three classes of basis elements with interdependent values of traces, which may be otherwise arbitrary. Representing the basis by four pairs of well-matched parentheses and denoting $k$ nested parentheses by $k$, these classes are

```
4
31, 22, 13, 211, 112, 1111
121, (12), (21), (11)1, 1(11), (111), ((11))
```

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