Preprint
Sliced skein algebras and geometric Kauffman bracket
arXiv.org
Cornell University Library, arXiv.org
10/09/2023
DOI: 10.48550/arxiv.2310.06189
Abstract
The sliced skein algebra of a closed surface of genus \(g\) with \(m\) punctures, \(\fS=\Sigma_{g,m}\), is the quotient of the Kauffman bracket skein algebra \(\SxS\) corresponding to fixing the scalar values of its peripheral curves. We show that the sliced skein algebra of a finite type surface is a domain if the ground ring is a domain. When the quantum parameter \(\xi\) is a root of unity we calculate the center of the sliced skein algebra and its PI-degree. Among applications we show that any smooth point of a sliced character variety is a fully Azumaya point of the skein algebra \(\SxS\). For any \(SL_2(\BC)\)--representation \(\rho\) of the fundamental group of an oriented connected 3-manifold \(M\) and a root of unity \(\xi\) with odd \(\ord(\xi^2)\), we introduce the \(\rho\)-reduced skein module \(\cS_{\xi, \rho}(M)\). We show that \(\cS_{\xi, \rho}(M)\) has dimension 1 when \(M\) is closed and \(\rho\) is irreducible. We also show that if \(\rho\) is irreducible the \(\rho\)-reduced skein module of a handlebody, as a module over the skein algebra of its boundary, is simple and has the dimension equal to the PI-degree of the skein algebra of its boundary.
Details
- Title: Subtitle
- Sliced skein algebras and geometric Kauffman bracket
- Creators
- Charles FrohmanJoanna Kania-BartoszynskaThang Lê
- Resource Type
- Preprint
- Publication Details
- arXiv.org
- DOI
- 10.48550/arxiv.2310.06189
- eISSN
- 2331-8422
- Publisher
- Cornell University Library, arXiv.org; Ithaca
- Language
- English
- Date posted
- 10/09/2023
- Academic Unit
- Mathematics
- Record Identifier
- 9984476556602771
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