arXiv subjects

Gad Naot

Publications and source records attributed to Gad Naot.

3 recordsLinked to original sources

arXiv2009

The universal Khovanov link homology theory

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.

math.GT↗

arXiv2007

The Universal sl_2 Link Homology Theory

We explore the complex associated to a link in the geometric formalism of Khovanov's (n=2) link homology theory, determine its exact underlying algebraic structure and find its precise universality properties for link homology functors. We present new methods of extracting all known link homology theories directly from this universal complex, and determine its relative strength as a link invariant by specifying the amount of information held within the complex. We achieve these goals by finding a complex isomorphism which reduces the complex into one in a simpler category. We introduce few tools and methods, including surface classification modulo the 4TU/S/T relations and genus generating operators, and use them to explore the relation between the geometric complex and its underlying algebraic structure. We identify the universal topological quantum field theory (TQFT) that can be used to create link homology and find that it is ``smaller'' than what was previously reported by Khovanov. We find new homology theories that hold a controlled amount of information relative to the known ones. The universal complex is computable efficiently using our reduction theorem. This allows us to explore the phenomenological aspects of link homology theory through the eyes of the universal complex in order to explain and unify various phenomena (such as torsion and thickness). The universal theory also enables us to state results regarding specific link homology theories derived from it. The methods developed in this thesis can be combined with other known techniques (such as link homology spectral sequences) or used in the various extensions of Khovanov link homology (such as sl_3 link homology).

math.GT↗

arXiv2005

On Chern-Simons theory with an inhomogeneous gauge group and BF theory knot invariants

We study the Chern-Simons topological quantum field theory with an inhomogeneous gauge group, a non-semi-simple group obtained from a semi-simple one by taking its semi-direct product with its Lie algebra. We find that the standard knot observables (i.e. traces of holonomies along knots) essentially vanish, but yet, the non-semi-simplicity of our gauge group allows us to consider a class of un-orthodox observables which breaks gauge invariance at one point and which lead to a non-trivial theory on long knots in $\mathbb{R}^3$. We have two main morals : 1. In the non-semi-simple case, there is more to observe in Chern-Simons theory! There might be other interesting non semi-simple gauge groups to study in this context beyond our example. 2. In our case of an inhomogeneous gauge group, we find that Chern-Simons theory with the un-orthodox observable is actually the same as 3D BF theory with the Cattaneo-Cotta-Ramusino-Martellini knot observable. This leads to a simplification of their results and enables us to generalize and solve a problem they posed regarding the relation between BF theory and the Alexander-Conway polynomial. Our result is that the most general knot invariant coming from pure BF topological quantum field theory is in the algebra generated by the coefficients of the Alexander-Conway polynomial.

math.GT↗