Generalised Legendrian racks of Legendrian links
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866911207807516672 |
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| author | Karmakar, Biswadeep Saraf, Deepanshi Singh, Mahender |
| author_facet | Karmakar, Biswadeep Saraf, Deepanshi Singh, Mahender |
| contents | A generalised Legendrian rack is a rack equipped with a Legendrian structure, which is a pair of maps encoding the information of Legendrian Reidemeister moves together with up and down cusps in the front diagram of an oriented Legendrian link. Employing a purely rack theoretic approach, we associate a generalised Legendrian rack (or a GL-rack) to an oriented Legendrian link, and prove that it is an invariant under Legendrian isotopy. As immediate applications, we prove that this invariant distinguishes infinitely many oriented Legendrian unknots and oriented Legendrian trefoils. To comprehend their algebraic structure, we prove that every GL-rack admits a homogeneous representation. Further, using the idea of trunks, we define modules over GL-racks, and prove the equivalence of the category of GL-rack modules and the category of Beck modules over a fixed GL-rack. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_06854 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalised Legendrian racks of Legendrian links Karmakar, Biswadeep Saraf, Deepanshi Singh, Mahender Geometric Topology Quantum Algebra Primary 57K10, 57K12, Secondary 57K33 A generalised Legendrian rack is a rack equipped with a Legendrian structure, which is a pair of maps encoding the information of Legendrian Reidemeister moves together with up and down cusps in the front diagram of an oriented Legendrian link. Employing a purely rack theoretic approach, we associate a generalised Legendrian rack (or a GL-rack) to an oriented Legendrian link, and prove that it is an invariant under Legendrian isotopy. As immediate applications, we prove that this invariant distinguishes infinitely many oriented Legendrian unknots and oriented Legendrian trefoils. To comprehend their algebraic structure, we prove that every GL-rack admits a homogeneous representation. Further, using the idea of trunks, we define modules over GL-racks, and prove the equivalence of the category of GL-rack modules and the category of Beck modules over a fixed GL-rack. |
| title | Generalised Legendrian racks of Legendrian links |
| topic | Geometric Topology Quantum Algebra Primary 57K10, 57K12, Secondary 57K33 |
| url | https://arxiv.org/abs/2301.06854 |