Decompositions of augmentation varieties via weaves and rulings
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912557537689600 |
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| author | Asplund, Johan Capovilla-Searle, Orsola Hughes, James Leverson, Caitlin Li, Wenyuan Wu, Angela |
| author_facet | Asplund, Johan Capovilla-Searle, Orsola Hughes, James Leverson, Caitlin Li, Wenyuan Wu, Angela |
| contents | The braid variety of a positive braid and the augmentation variety of a Legendrian link both admit decompositions coming from weaves and rulings, respectively. We prove that these decompositions agree under an isomorphism between the braid variety and the augmentation variety. We also prove that both decompositions coincide with a Deodhar decomposition and another decomposition coming from the microlocal theory of sheaves. Our proof relies on a detailed comparison between weaves and Morse complex sequences. Among other things, we show that the cluster variables of the maximal cluster torus of the augmentation variety can be computed from the Legendrian via Morse complex sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decompositions of augmentation varieties via weaves and rulings Asplund, Johan Capovilla-Searle, Orsola Hughes, James Leverson, Caitlin Li, Wenyuan Wu, Angela Symplectic Geometry Algebraic Geometry Combinatorics Representation Theory The braid variety of a positive braid and the augmentation variety of a Legendrian link both admit decompositions coming from weaves and rulings, respectively. We prove that these decompositions agree under an isomorphism between the braid variety and the augmentation variety. We also prove that both decompositions coincide with a Deodhar decomposition and another decomposition coming from the microlocal theory of sheaves. Our proof relies on a detailed comparison between weaves and Morse complex sequences. Among other things, we show that the cluster variables of the maximal cluster torus of the augmentation variety can be computed from the Legendrian via Morse complex sequences. |
| title | Decompositions of augmentation varieties via weaves and rulings |
| topic | Symplectic Geometry Algebraic Geometry Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2508.20226 |