Knot Floer Homology, the Burau Representation, and Quantum $\mathfrak{gl}(1 \vert 1)$

TL;DR

The paper builds a concrete geometric bridge between Burau/Gassner representations and knot Floer theory by constructing the triply-graded invariant $\widehat{HFB}$ that categorifies determinants of submatrices of $\psi_n(\rho)-\lambda I_n$, and extends these ideas to bordered sutured Floer theory for tangles. It then connects these Floer-theoretic invariants to quantum $U_q(\mathfrak{gl}(1|1))$ via wedge representations and state-sum formulations, showing that decategorified data recovers Alexander polynomials while a bordered approach recovers the full braiding representation on tensor powers. The work provides a transparent, geometric route to relate knot Floer homology with quantum group invariants and offers a decategorified tangle invariant $\Delta_{T,\mathbf{j},\mathbf{k}}$, tying together Heegaard Floer theory, Fox calculus, and Reshetikhin–Turaev-type invariants. Overall, the paper advances a coherent framework uniting low-dimensional topology, braid group representations, and quantum algebra with potential for broad generalizations beyond the braid setting. The constructions yield new braid invariants, clarify the role of Alexander-type polynomials in HF contexts, and give a practical geometric pathway to the $U_q(\mathfrak{gl}(1|1))$ braid representation via bordered sutured Floer theory.

Abstract

The Burau representation of braid groups and knot Floer homology share a link to the Fox calculus. We make this connection explicit, with the following outcome: if $B$ is the full Burau matrix of any braid, and $A$ is any square submatrix of $B - λI$, we define a Heegaard Floer homology theory that categorifies $\det(A)$ and is an invariant of the braid. We also describe an analogous construction for the Gassner representation. Then, we leverage the relationship between the Burau representation and quantum $\mathfrak{gl}(1 \vert 1)$ to exhibit connections between the latter and Heegaard Floer homology. We associate a bordered sutured Heegaard Floer homology group to any tangle, and give a simple, geometric proof that our invariant recovers the $U_q(\mathfrak{gl}(1 \vert 1))$ braid representation.

Figures (7)

• Figure 1:
• Figure 2: The bridge diagram $\mathcal{B}(\rho)$ for the braid $\rho = \sigma_1^2 \in B_2$
• Figure 3: Two Heegaard diagrams for the braid $\rho = \sigma_1^2 \in B_2$. The labelled circles indicate handle attachments (and compare Figure \ref{['fig:bridge_diag']}).
• Figure 4: Comparing chopsticks and noodles with $\alpha$ and $\beta$ curves
• Figure 5: Using $\rho$ to express $K$ as a $(1,1)$-tangle

Theorems & Definitions (74)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 3.1
• Remark 3.2
• Definition 3.3
• Remark 3.4
• Definition 3.5
• Lemma 3.6
• Lemma 3.7