Sliced skein algebras and geometric Kauffman bracket
Charles Frohman, Joanna Kania-Bartoszynska, Thang L\^e

TL;DR
This paper studies sliced skein algebras of surfaces, calculating their centers and PI-degrees at roots of unity, and explores their properties and modules related to 3-manifold representations, revealing new algebraic and geometric insights.
Contribution
It establishes that sliced skein algebras are domains, computes their centers and PI-degrees at roots of unity, and introduces rho-reduced skein modules with applications to 3-manifold representations.
Findings
Sliced skein algebra of a finite type surface is a domain if the ground ring is a domain.
Centers and PI-degrees of sliced skein algebras are computed at roots of unity.
rho-reduced skein modules have dimension 1 for closed manifolds with irreducible representations.
Abstract
The sliced skein algebra of a closed surface of genus with punctures, , is the quotient of the Kauffman bracket skein algebra 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 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 . For any --representation of the fundamental group of an oriented connected 3-manifold and a root of unity with odd , we introduce the -reduced skein module . We show that…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Algebraic structures and combinatorial models · Advanced Operator Algebra Research
