Knot theory and cluster algebras
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866910430573625344 |
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| author | Bazier-Matte, Véronique Schiffler, Ralf |
| author_facet | Bazier-Matte, Véronique Schiffler, Ralf |
| contents | We establish a connection between knot theory and cluster algebras via representation theory. To every knot diagram (or link diagram), we associate a cluster algebra by constructing a quiver with potential. The rank of the cluster algebra is $2n$, where $n$ is the number of crossing points in the knot diagram. We then construct $2n$ indecomposable modules $T(i)$ over the Jacobian algebra of the quiver with potential. For each $T(i)$, we show that the submodule lattice is isomorphic to the corresponding lattice of Kauffman states. We then give a realization of the Alexander polynomial of the knot as a specialization of the $F$-polynomial of $T(i)$, for every $i$. Furthermore, we conjecture that the collection of the $T(i)$ forms a cluster in the cluster algebra whose quiver is isomorphic to the opposite of the initial quiver, and that the resulting cluster automorphism is of order two. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_14740 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Knot theory and cluster algebras Bazier-Matte, Véronique Schiffler, Ralf Representation Theory Combinatorics General Topology 13F60, 57K14, 16G20 We establish a connection between knot theory and cluster algebras via representation theory. To every knot diagram (or link diagram), we associate a cluster algebra by constructing a quiver with potential. The rank of the cluster algebra is $2n$, where $n$ is the number of crossing points in the knot diagram. We then construct $2n$ indecomposable modules $T(i)$ over the Jacobian algebra of the quiver with potential. For each $T(i)$, we show that the submodule lattice is isomorphic to the corresponding lattice of Kauffman states. We then give a realization of the Alexander polynomial of the knot as a specialization of the $F$-polynomial of $T(i)$, for every $i$. Furthermore, we conjecture that the collection of the $T(i)$ forms a cluster in the cluster algebra whose quiver is isomorphic to the opposite of the initial quiver, and that the resulting cluster automorphism is of order two. |
| title | Knot theory and cluster algebras |
| topic | Representation Theory Combinatorics General Topology 13F60, 57K14, 16G20 |
| url | https://arxiv.org/abs/2110.14740 |