  We determine the algebraic structure underlying the geometric complex
associated to a link in Bar-Natan's geometric formalism of Khovanov's link
homology theory (n=2). We find an isomorphism of complexes which reduces the
complex to one in a simpler category. This reduction enables us to specify
exactly the amount of information held within the geometric complex and thus
state precisely its universality properties for link homology theories. We also
determine its strength as a link invariant relative to the different
topological quantum field theories (TQFTs) used to create link homology. We
identify the most general (universal) TQFT that can be used to create link
homology and find that it is "smaller" than the TQFT previously reported by
Khovanov as the universal link homology theory. We give a new method of
extracting all other link homology theories (including Khovanov's universal
TQFT) directly from the universal geometric complex, along with new homology
theories that hold a controlled amount of information. We achieve these goals
by making a classification of surfaces (with boundaries) modulo the 4TU/S/T
relations, a process involving the introduction of genus generating operators.
These operators enable us to explore the relation between the geometric complex
and its algebraic structure.
