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
(). 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.
Keywords
categorification, cobordism, Jones polynomial, Khovanov
link homology, quantum knot invariants, TQFT
