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On Kauffman bracket skein algebras of marked surfaces and the Chebyshev-Frobenius homomorphism
by
Jonathan Paprocki
Georgia Tech
Coauthors: Thang T. Q. Le
Kauffman bracket skein algebras of marked surfaces admit both knots and arcs ending at marked points. Muller showed that skein algebras of marked surfaces embed into a very simple algebra called a quantum torus.
We show that the Chebyshev homomorphism of Bonahon and Wong between unmarked surface skein algebras at certain roots of unity is induced by a sort of Frobenius homomorphism between quantum torii obtained by marking the surface. We use this technique to extend their result to define a "Chebyshev-Frobenius" linear transformation between skein modules of (un)marked 3-manifolds at certain roots of unity.
In addition, we extend Muller's result to allow marked surfaces with unmarked boundary components. This allows us to define a surgery theory, relating skein algebras between surfaces obtained by plugging holes of boundary components and adding/removing marked points.
Date received: March 7, 2016
Copyright © 2016 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbmk-12.