Knot Floer Homology, the Burau Representation, and Quantum $\mathfrak{gl}(1 \vert 1)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909839331950592 |
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| author | Boninger, Joe |
| author_facet | Boninger, Joe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15321 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Knot Floer Homology, the Burau Representation, and Quantum $\mathfrak{gl}(1 \vert 1)$ Boninger, Joe Geometric Topology Quantum Algebra 57K10, 57K16, 57K18 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. |
| title | Knot Floer Homology, the Burau Representation, and Quantum $\mathfrak{gl}(1 \vert 1)$ |
| topic | Geometric Topology Quantum Algebra 57K10, 57K16, 57K18 |
| url | https://arxiv.org/abs/2509.15321 |