Knot Floer Homology, the Burau Representation, and Quantum \mathfrakgl(1 \vert 1)

2025/09/18 by Joe Boninger, Boninger, Joe
Mathematics · #57K10 #57K16 #57K18 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2509.15321

openalex publication_date 2025/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \mathfrakgl(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(\mathfrakgl(1 \vert 1)) braid representation.

Citations

• Kauffman states, bordered algebras, and a bigraded knot invariant
• Categorified invariants and the braid group
• Catégorification des coefficients de la matrice de la représentation de Burau
• Bordered Heegaard Floer homology
• A combinatorial description of knot Floer homology
• Knot Floer homology detects genus-one fibred knots
• Holomorphic disks and topological invariants for closed three-manifolds
• Holomorphic disks and topological invariants for closed three-manifolds
• Does the Jones polynomial detect the unknot?
• Braid Groups are Linear
• The Gassner representation for string links
• The Multivariable Alexander Polynomial for a Closed Braid

Related

• Trivalent vertices and bordered knot Floer homology in the standard basis
• Homological Link Invariants from Floer Theory
• Braids and combinatorial knot Floer homology
• Garside-theoretic analysis of Burau representations
• Schur-Weyl type duality for quantized gl(1|1),the Burau representation of braid groups and invariants of tangled graphs