26 research outputs found
Skein Modules from Skew Howe Duality and Affine Extensions
We show that we can release the rigidity of the skew Howe duality process for
knot invariants by rescaling the quantum Weyl group action,
and recover skein modules for web-tangles. This skew Howe duality phenomenon
can be extended to the affine case, corresponding to looking
at tangles embedded in a solid torus. We investigate the relations between the
invariants constructed by evaluation representations (and affinization of them)
and usual skein modules, and give tools for interpretations of annular skein
modules as sub-algebras of intertwiners for particular
representations. The categorification proposed in a joint work with A. Lauda
and D. Rose also admits a direct extension in the affine case
Chebyshev polynomials and the Frohman-Gelca formula
Using Chebyshev polynomials, C. Frohman and R. Gelca introduce a basis of the
Kauffman bracket skein module of the torus. This basis is especially useful
because the Jones-Kauffman product can be described via a very simple
Product-to-Sum formula. Presented in this work is a diagrammatic proof of this
formula, which emphasizes and demystifies the role played by Chebyshev
polynomials.Comment: 13 page
Gl2 Foam Functoriality and Skein Positivity
We prove full functoriality of Khovanov homology for tangled framed gl2 webs.
We use this functoriality result to prove a strong positivity result for
(orientable) surface skein algebras. The argument goes categorical and consists
in proving that so-called linear complexes are stable under superposition.Comment: 71 pages. Preliminary version, comments welcome