  This thesis studies the Bar-Natan skein module of the solid torus with a
particular boundary curve system, and in particular a diagrammatic presentation
of it due to Russell. This module has deep connections to topology and
categorification: it is isomorphic to both the total homology of the
(n,n)-Springer variety and the 0th Hochschild homology of the Khovanov arc ring
H^n. We can also view the Bar-Natan--Russell skein module from a
representation-theoretic viewpoint as an extension of the Frenkel--Khovanov
graphical description of the Lusztig dual canonical basis of the nth tensor
power of the fundamental U_q(sl_2)-representation. One of our primary results
is to extend a dualization construction of Khovanov using Jones--Wenzl
projectors from the Lusztig basis to the Russell basis. We also construct and
explore several deformations of the Russell skein module. One deformation is a
quantum deformation that arises from embedding the Russell skein module in a
space that obeys Kauffman--Lins diagrammatic relations. Our quantum version
recovers the original Russell space when q is specialized to -1 and carries a
natural braid group action that recovers the symmetric group action of Russell
and Tymoczko. We also present an equivariant deformation that arises from
replacing the TQFT algebra A used in the construction of the rings H^n by the
equivariant homology of the two-sphere with the standard action of U(2) and
taking the 0th Hochschild homology of the resulting deformed arc rings. We show
that the equivariant deformation has the expected rank. Finally, we consider
the Khovanov two-functor F from the category of tangles. We show that it
induces a surjection from the space of cobordisms of planar (2m, 2n)-tangles to
the space of (H^m, H^n)-bimodule homomorphisms and give an explicit description
of the kernel. We use our result to introduce a new quotient of the Russell
skein module.
