Quantum traces for $\mathrm{SL}_n(\mathbb{C})$: The case $n=3$
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866914788669390848 |
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| author | Douglas, Daniel C. |
| author_facet | Douglas, Daniel C. |
| contents | We generalize Bonahon-Wong's $\mathrm{SL}_2(\mathbb{C})$-quantum trace map to the setting of $\mathrm{SL}_3(\mathbb{C})$. More precisely, given a non-zero complex parameter $q=e^{2 πi \hbar}$, we associate to each isotopy class of framed oriented links $K$ in a thickened punctured surface $\mathfrak{S} \times (0, 1)$ a Laurent polynomial $\mathrm{Tr}_λ^q(K) = \mathrm{Tr}_λ^q(K)(X_i^q)$ in $q$-deformations $X_i^q$ of the Fock-Goncharov $\mathcal{X}$-coordinates for higher Teichmüller space. This construction depends on a choice $λ$ of ideal triangulation of the surface $\mathfrak{S}$. Along the way, we propose a definition for a $\mathrm{SL}_n(\mathbb{C})$-version of this invariant. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2101_06817 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quantum traces for $\mathrm{SL}_n(\mathbb{C})$: The case $n=3$ Douglas, Daniel C. Geometric Topology Quantum Algebra 2020: 57K31, 32G15, 20G42 We generalize Bonahon-Wong's $\mathrm{SL}_2(\mathbb{C})$-quantum trace map to the setting of $\mathrm{SL}_3(\mathbb{C})$. More precisely, given a non-zero complex parameter $q=e^{2 πi \hbar}$, we associate to each isotopy class of framed oriented links $K$ in a thickened punctured surface $\mathfrak{S} \times (0, 1)$ a Laurent polynomial $\mathrm{Tr}_λ^q(K) = \mathrm{Tr}_λ^q(K)(X_i^q)$ in $q$-deformations $X_i^q$ of the Fock-Goncharov $\mathcal{X}$-coordinates for higher Teichmüller space. This construction depends on a choice $λ$ of ideal triangulation of the surface $\mathfrak{S}$. Along the way, we propose a definition for a $\mathrm{SL}_n(\mathbb{C})$-version of this invariant. |
| title | Quantum traces for $\mathrm{SL}_n(\mathbb{C})$: The case $n=3$ |
| topic | Geometric Topology Quantum Algebra 2020: 57K31, 32G15, 20G42 |
| url | https://arxiv.org/abs/2101.06817 |