On skein algebras of planar surfaces
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866911614241865728 |
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| author | Chen, Haimiao |
| author_facet | Chen, Haimiao |
| contents | Let $R$ be a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$.
Let $\mathcal{S}_n$ denote the Kauffman bracket skein algebra of the $n$-holed disk $Σ_{0,n+1}$ over $R$.
When $q+q^{-1}$ is invertible, in 2000 Przytycki and Sikora found a set of $n+{n\choose 2}+{n\choose 3}$ generators for $\mathcal{S}_n$;
we show that the ideal of defining relations among these generators is generated by relations of degree $\le6$ supported by certain subsurfaces diffeomorphic to $Σ_{0,k+1}$ with $k\le 6$. When $q+q^{-1}$ is not invertible, a set of $2^n-1$ generators for $\mathcal{S}_n$ was known to Bullock in 1999; we show that the ideal of defining relations is generated by relations of degree $\le 2k+2$ supported by certain subsurfaces diffeomorphic to $Σ_{0,k+1}$ with $k\le n$. These results are substantial progresses towards answering Problem 1.92 (J) in the Kirby's list. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_07856 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On skein algebras of planar surfaces Chen, Haimiao Geometric Topology Quantum Algebra 57K16, 57K31 Let $R$ be a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$. Let $\mathcal{S}_n$ denote the Kauffman bracket skein algebra of the $n$-holed disk $Σ_{0,n+1}$ over $R$. When $q+q^{-1}$ is invertible, in 2000 Przytycki and Sikora found a set of $n+{n\choose 2}+{n\choose 3}$ generators for $\mathcal{S}_n$; we show that the ideal of defining relations among these generators is generated by relations of degree $\le6$ supported by certain subsurfaces diffeomorphic to $Σ_{0,k+1}$ with $k\le 6$. When $q+q^{-1}$ is not invertible, a set of $2^n-1$ generators for $\mathcal{S}_n$ was known to Bullock in 1999; we show that the ideal of defining relations is generated by relations of degree $\le 2k+2$ supported by certain subsurfaces diffeomorphic to $Σ_{0,k+1}$ with $k\le n$. These results are substantial progresses towards answering Problem 1.92 (J) in the Kirby's list. |
| title | On skein algebras of planar surfaces |
| topic | Geometric Topology Quantum Algebra 57K16, 57K31 |
| url | https://arxiv.org/abs/2206.07856 |