Link invariants from Hecke relations of higher order

Question

Alexander theorem says oriented links in $\mathbb{R}^3$ can be represented by closures of braids. Markov theorem says that braids related by Markov moves produce isotopic braid closures, and vice versa. Hence any map $\Phi$ with domain the infinite braid group $B_\infty$ (or its group algebra) that equalizes on the Markov relations gives an invariant of oriented links.

For example, in [1], Vaughan Jones showed that with $\Phi = Tr \circ Pr$, where $Tr$ the (rescaled) Ocneanu trace and $Pr$ the quotient map to the Hecke algebra of type A, the HOMFLY-PT polynomial is recovered. This 2-variated polynomial is very strong: It specializes to many other well-known link invariants including the Alexander polynomial and the Jones polynomial. With the Hecke algebra being semisimple (generic $q$), Artin-Wedderburn decomposes it into a direct product of matrix algebras. Jones then showed that the Ocneanu trace $Tr$ is just the matrix trace suitably scaled on each multiplicand. (In fact, the Jones polynomial can also be obtained with a similar fashion by further quotienting the Hecke algebra to the TL algebra [2].)

Question

$Pr$ passes the braid group algebra (infinite dimensional) to the easier Hecke algebra (finite dimensional) and may potentially lose information. Are there any known link invariants arise in similar fashions, but do not "factor through" Hecke algebra?

For example, instead of quotienting the quadratic relation $\sigma^{2} = (q-1) \sigma + q$, why not try relations of higher order (cubic, quartic, so on)? Or, how about not quotienting (thus not losing any information) at all? It must be a difficult approach, but I hope to see some efforts made in this direction.

Related

• From braid representations to link invariants

Reference

• [1] Hecke Algebra Representations of Braid Groups and Link by V. F. R. Jones
• [2] Local unitary representations of the braid group and their applications to quantum computing by Colleen Delaney, Eric Rowell, and Zhenghan Wang

Answer 1

Since you've referenced the paper by Delaney, Rowell and Wang (where the authors talk about braided fusion categories), I'll add a comment (in response to your question about taking more interesting quotients) on how 'higher order' Hecke algebras naturally appear in unitary braided fusion categories, and how tracing then yields framed link invariants.

Definition: Let's define a (generalised) Iwahori-Hecke algebra $H_{n}(Q,k)$ to be the quotient of braid algebra $\mathbb{C}[B_{n}]$ by an ideal generated by $\{\Pi_{i=1}^{k}(\sigma_j - r_{i})\}_{j=1}^{n-1}$ where $r_{i}\in\mathbb{C}^{\times}$ are fixed, and $k\geq2$. The case $k=2$ corresponds to the usual Iwahori-Hecke algebra.

Let $\mathcal{C}$ be a unitary braided fusion category, and $Irr(\mathcal{C})$ denote a set of representatives of isomorphism classes of simple objects in $\mathcal{C}$. Take some self-dual $q\in Irr(\mathcal{C})$ such that $q\otimes q \cong \bigoplus_{i=1}^{k} x_{i}$ where $k\geq 2$ and the simple subobjects $\{x_{i}\}_{i}$ are all distinct elements of $Irr(\mathcal{C})$. It turns out that the action of the $n$-strand braid group $B_{n}$ on $End(q^{\otimes n})$ defines a unitary representation $\rho$ of the generalised Iwahori-Hecke algebra $H_{n}(Q, k)$, where $\{r_{i}\}_{i=1}^{k}$ are the eigenvalues of the unitary braiding matrix $R^{qq}\in End(q\otimes q)$. If you now take an arbitrary $n$-braid $f_{n} \in End(q^{\otimes n})$ and perform the (quantum) trace $\widetilde{Tr}$, you will have a link diagram that evaluates to a scalar $\widetilde{Tr}(f_{n})\in\mathbb{C}$ which is invariant under braid isotopy. (N.B. the quantum trace can be understood as a weighted matrix trace, see attached notes).

In the above, the framed link invariant corresponding to the case $k=2$ is the Kauffman bracket. For $k=3$, it's the Dubrovnik or Kauffman polynomial -- these can be seen as coming from the 'cubic' Hecke algebra $H_{n}(Q,3)$. In both cases, the representation $\rho$ will also factor through the Temperley-Lieb algebra. Check out the paper [1] (specifically Theorem 3.1) by Morrison, Peters and Snyder. However, I'm not sure if there are results on classifying invariants this way for $k>3$.$^{\dagger}$

Attachment: In response to OP's request, rough notes explaining how representations of the Iwahori-Hecke algebra arise in unitary braided fusion categories can be found at [2].

[1] https://arxiv.org/abs/1003.0022

[2] https://sites.google.com/view/sachinvalera/rough-notes

$\dagger $ A partial result for $k=4$ is claimed (yet to appear?) in [1] that corresponds to Kuperberg's G2 invariant.

Answer 2

For the Kauffman polynomial, the Birman-Murakami-Wenzl algebra (BMW algebra) https://en.wikipedia.org/wiki/Birman%E2%80%93Wenzl_algebra plays a similar role as Hecke algebra.

As an algebra, the BMW algebla is obtained from the group algebra of braid groups by cubic relation and some additional relations. The BMW algebra has the trace function $Tr$, and like the Hecke algebra formulaion of the HOMFLY-PT polynomial, the Kauffman polynomial is obtianed by exatly the the same manner, $\mbox{Kauffman polynomial} = Tr \circ Pr$ where $Pr$ is the quotient map (with suitable renormalization).