# Kauffman's state model for the Alexander polynomial, via representation theory

I've been reading Oleg Viro's paper on "quantum relatives of the Alexander polynomial" (arXiv:math/0204290), which, among other more general things, derives state-sum formulas for the Alexander polynomial of a knot in terms of the representation theory of $\mathcal{U}_q(\mathfrak{gl}_{1|1})$ and (alternatively) $\mathcal{U}_q(\mathfrak{sl}_2)$ with $q$ a fourth root of unity.

But the state-sum model I know and love for the Alexander polynomial is Kauffman's original one (from his book "Formal Knot Theory"). Is this also derived from representation theory, like in Viro's paper (for one of these algebras or some other one)?

I'm interested in this question because the Kauffman state model is useful in knot Floer homology, which categorifies the Alexander polynomial. As far as I know (e.g. Sartori's recent preprint arXiv:1305.6162), a representation-theoretic categorification of the Alexander polynomial, thought of as a quantum $\mathfrak{gl}_{1|1}$-invariant, doesn't yet exist but may exist soon. I'd be very curious to know what sort of representation-theoretic structures knot Floer homology corresponds to, but as this seems to be a more difficult question, I'll stick with the Kauffman states for now!

-

Update: I've found at least a partial answer to this question, so I thought I'd write it up here and see if it's familiar, or if anyone can continue the story.

Kauffman's state sum model from "Formal Knot Theory" admits a generalization for tangles. I got this from Ozsvath and Szabo's new "bordered" construction of knot Floer homology, but I'm not sure whether this bit itself is new or whether someone's written it down before (I'd be curious to know). It goes as follows (restricted to braids for simplicity here, although it works in more generality):

To $n$ points on the $x$ axis, associate a vector space $\mathcal{I_n}$ of dimension $2^{n-1}$ over $\mathbb{C}(t)$. The space $\mathcal{I_n}$ can be naturally associated with $\wedge^*(R)$, where $R$ is the vector space of dimension $n-1$ generated by the line-intervals $r_1, \ldots, r_{n-1}$ between the $n$ points. A generator $r_{i_1} \wedge \ldots \wedge r_{i_k}$ is depicted by drawing $n$ small vertical lines over the $n$ points, creating $n-1$ small regions, and then putting dots in the regions corresponding to $r_{i_1}$ through $r_{i_k}$.

To a braid $B$ with $n$ strands, associate a map $f(B): \mathcal{I_n} \to \mathcal{I_n}$ defined by summing over "partial Kauffman states." These are assignments of a dot to one corner of each crossing of the braid such that no planar region has more than one dot. A partial Kauffman state $\sigma$ is compatible with a generator $r = r_{i_1} \wedge \ldots \wedge r_{i_k}$ of $\mathcal{I}_n$ on the input side (say the top of the braid) if all planar regions, adjacent to the top boundary, which are assigned dots in $\sigma$, are included among the regions $r_{i_1}, \ldots, r_{i_k}$. Then $f(B)$, applied to $r$, is a sum over all partial Kauffman states compatible with it. Each partial Kauffman state $\sigma$ determines a coefficient $c(\sigma)$ in $\mathbb{C}(t)$ (as a product over local contributions from corners) and another generator $r'$ of $\mathcal{I}_n$. The generator $r'$ has a region $r_i$ if either the corresponding planar region has no dot and does not intersect the top boundary, or the planar region intersects both boundaries, has no dot from $\sigma$, and $r_i$ is among the $r_{i_1}, \ldots, r_{i_k}$. Then we can write $f(B)(r) = \sum_{\sigma} c(\sigma) \cdot r'$, where the sum is over $\sigma$ compatible with $r$.

Surprisingly (to me), the data $\mathcal{I}_n$ and $f(B)$ can be related to quantum groups. Let $V$ denote the vector representation of $\mathcal{U}_q(\mathfrak{gl}_{1|1})$, with basis $\{v,w\}$ such that $|v| = 0$ and $|w| = 1$. Then, as a vector space over $\mathbb{C}(q)$, the tensor product $V^{\otimes n}$ can be identified with $\wedge^*(E)$, where $E$ is the $n$-dimensional vector space spanned by $\{w \otimes v \otimes \ldots \otimes v, v \otimes w \otimes \ldots \otimes v, v \otimes \ldots \otimes w\}$. Call these elements $e_1, \ldots, e_n$. Then consider the $n-1$-dimensional subspace $E_0$ of $E$, spanned by $\{e_1 - q^{-1}e_2, e_2 - q^{-1} e_3, \ldots, e_{n-1} - q^{-1} e_n\}$. The exterior product $\wedge^*(E_0)$ is a $2^{n-1}$-dimensional subspace of $V^{\otimes n}$ which is preserved by the action of the braid group via R-matrices. Furthermore, the braid group action on $V^{\otimes n}$ restricted to this subspace coincides with the map $f(B)$ under the identification $r_i \leftrightarrow (e_i - q^{-1} e_{i+1})$ and $t \leftrightarrow q^2$ (this is a computation which I didn't write here because this post is already long).

I'm very interested in understanding this connection more fully. As far as I can tell, the elements $e_i - q^{-1}e_{i+1}$ are part of the Lusztig canonical basis of $V^{\otimes n}$ at $q = \infty$ rather than $q = 0$. Is there a better way to describe these basis elements and the half-dimensional subspace they span? Does this picture have analogues for other groups (particularly $\mathfrak{sl}_2$)? And, of course, if these things are discussed in a paper somewhere, I'd be very glad to hear about it.

-