Representations of knot groups in $\textrm{AGL}_{1}(\mathbb{C})$ and Alexander invariants
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916971028676608 |
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| author | González-Prieto, Ángel Martínez, Javier Muñoz, Vicente |
| author_facet | González-Prieto, Ángel Martínez, Javier Muñoz, Vicente |
| contents | This paper reinterprets Alexander-type invariants of knots via representation varieties of knot groups into the group $\textrm{AGL}_1(\mathbb{C})$ of affine transformations of the complex line. In particular, we prove that the coordinate ring of the $\textrm{AGL}_{1}(\mathbb{C})$-representation variety is isomorphic to the symmetric algebra of the Alexander module. This yields a natural interpretation of the Alexander polynomial as the singular locus of a coherent sheaf over $\mathbb{C}^*$, whose fibres correspond to quandle representation varieties of the knot quandle. As a by-product, we construct Topological Quantum Field Theories that provide effective computational methods and recover the Burau representations of braids. This theory offers a new geometric perspective on classical Alexander invariants and their functorial quantization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23364 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representations of knot groups in $\textrm{AGL}_{1}(\mathbb{C})$ and Alexander invariants González-Prieto, Ángel Martínez, Javier Muñoz, Vicente Geometric Topology Quantum Algebra Representation Theory 57K14 (Primary), 14M35, 57K12, 18M20 (Secondary) This paper reinterprets Alexander-type invariants of knots via representation varieties of knot groups into the group $\textrm{AGL}_1(\mathbb{C})$ of affine transformations of the complex line. In particular, we prove that the coordinate ring of the $\textrm{AGL}_{1}(\mathbb{C})$-representation variety is isomorphic to the symmetric algebra of the Alexander module. This yields a natural interpretation of the Alexander polynomial as the singular locus of a coherent sheaf over $\mathbb{C}^*$, whose fibres correspond to quandle representation varieties of the knot quandle. As a by-product, we construct Topological Quantum Field Theories that provide effective computational methods and recover the Burau representations of braids. This theory offers a new geometric perspective on classical Alexander invariants and their functorial quantization. |
| title | Representations of knot groups in $\textrm{AGL}_{1}(\mathbb{C})$ and Alexander invariants |
| topic | Geometric Topology Quantum Algebra Representation Theory 57K14 (Primary), 14M35, 57K12, 18M20 (Secondary) |
| url | https://arxiv.org/abs/2503.23364 |