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This thesis studies the Bar-Natan skein module of the solid torus with aparticular boundary curve system, and in particular a diagrammatic presentationof it due to Russell. This module has deep connections to topology andcategorification: it is isomorphic to both the total homology of the(n,n)-Springer variety and the 0th Hochschild homology of the Khovanov arc ringH^n. We can also view the Bar-Natan--Russell skein module from arepresentation-theoretic viewpoint as an extension of the Frenkel--Khovanovgraphical description of the Lusztig dual canonical basis of the nth tensorpower of the fundamental U_q(sl_2)-representation. One of our primary resultsis to extend a dualization construction of Khovanov using Jones--Wenzlprojectors from the Lusztig basis to the Russell basis. We also construct andexplore several deformations of the Russell skein module. One deformation is aquantum deformation that arises from embedding the Russell skein module in aspace that obeys Kauffman--Lins diagrammatic relations. Our quantum versionrecovers the original Russell space when q is specialized to -1 and carries anatural braid group action that recovers the symmetric group action of Russelland Tymoczko. We also present an equivariant deformation that arises fromreplacing the TQFT algebra A used in the construction of the rings H^n by theequivariant homology of the two-sphere with the standard action of U(2) andtaking the 0th Hochschild homology of the resulting deformed arc rings. We showthat the equivariant deformation has the expected rank. Finally, we considerthe Khovanov two-functor F from the category of tangles. We show that itinduces a surjection from the space of cobordisms of planar (2m, 2n)-tangles tothe space of (H^m, H^n)-bimodule homomorphisms and give an explicit descriptionof the kernel. We use our result to introduce a new quotient of the Russellskein module.

Dualization and deformations of the Bar-Natan--Russell skein module (PhD thesis)