Augmentations, annuli, and Alexander polynomials
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866914707644874752 |
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| author | Diogo, Luís Ekholm, Tobias |
| author_facet | Diogo, Luís Ekholm, Tobias |
| contents | The augmentation variety of a knot is the locus, in the 3-dimensional coefficient space of the knot contact homology dg-algebra, where the algebra admits a unital chain map to the complex numbers. We explain how to express the Alexander polynomial of a knot in terms of the augmentation variety: it is the exponential of the integral of a ratio of two partial derivatives. The expression is derived from a description of the Alexander polynomial as a count of Floer strips and holomorphic annuli, in the cotangent bundle of Euclidean 3-space, stretching between a Lagrangian with the topology of the knot complement and the zero-section, and from a description of the boundary of the moduli space of such annuli with one positive puncture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2005_09733 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Augmentations, annuli, and Alexander polynomials Diogo, Luís Ekholm, Tobias Symplectic Geometry Geometric Topology 53D40, 53D42, 57K10, 57K14 The augmentation variety of a knot is the locus, in the 3-dimensional coefficient space of the knot contact homology dg-algebra, where the algebra admits a unital chain map to the complex numbers. We explain how to express the Alexander polynomial of a knot in terms of the augmentation variety: it is the exponential of the integral of a ratio of two partial derivatives. The expression is derived from a description of the Alexander polynomial as a count of Floer strips and holomorphic annuli, in the cotangent bundle of Euclidean 3-space, stretching between a Lagrangian with the topology of the knot complement and the zero-section, and from a description of the boundary of the moduli space of such annuli with one positive puncture. |
| title | Augmentations, annuli, and Alexander polynomials |
| topic | Symplectic Geometry Geometric Topology 53D40, 53D42, 57K10, 57K14 |
| url | https://arxiv.org/abs/2005.09733 |