Journal article
Sliced skein algebras and geometric Kauffman bracket
Advances in mathematics (New York. 1965), Vol.463, 110118
03/2025
DOI: 10.1016/j.aim.2025.110118
Abstract
The sliced skein algebra of a closed surface of genus g with m punctures, S=Σg,m, is the quotient of the Kauffman bracket skein algebra Sξ(S) 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 ξ 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 an Azumaya point of the skein algebra Sξ(S).
For any SL2(C)-representation ρ of the fundamental group of an oriented connected 3-manifold M and a root of unity ξ with the order of ξ2 odd, we introduce the ρ-reduced skein module Sξ,ρ(M). We show that Sξ,ρ(M) has dimension 1 when M is closed and ρ is irreducible. We also show that if ρ is irreducible the ρ-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 D. Frohman - Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USAJoanna Kania-Bartoszynska - U.S. National Science FoundationThang T.Q. Lê - Georgia Institute of Technology
- Resource Type
- Journal article
- Publication Details
- Advances in mathematics (New York. 1965), Vol.463, 110118
- DOI
- 10.1016/j.aim.2025.110118
- ISSN
- 0001-8708
- eISSN
- 1090-2082
- Publisher
- Elsevier Inc
- Grant note
- NSF at the National Science Foundation: DMS-2203255
The authors thank F. Bleher, F. Bonahon, F. Costantino, R. Kinser, E. Letellier and A. Sikora for helpful discussions. We also thank T. Yu for suggesting Proposition 4.3. This material is based upon work supported by the NSF grant DMS-2203255 (T.L.) and done while serving (J.K.B.) at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors presented the results of the current paper at various conferences, in particular at Joint Math Meetings in Denver in January 2020, at virtual JMM in January 2021, and at the "Quantum Topology and Geometry" conference in Paris in June 2022, and would like to thank the organizers for allowing us to present our work.
- Language
- English
- Date published
- 03/2025
- Academic Unit
- Mathematics
- Record Identifier
- 9984786448602771
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