| The work bv M. Khovanov in 2000 introduced a way to construct Homology Groups that categorify the unnormalized Jones polynomial for a link. His work has sparked a big deal of interest in recent years. A natural question was to find which other polynomials can be categorified. In 2004, L. Helme-Guizon and Y. Rong introduced a categorification for the chromatic polynomial for graphs, which is a one variable polynomial. The next natural polynomial to explore is the Tutte polynomial, a two-variable polynomial that contains more information than the chromatic polynomial. In this dissertation we introduce the construction of a cochain complex associated to graphs. The corresponding cohomology categorifies a version of the Tutte polynomial.; More specifically, given a graph G, we associate bigraded algebras to the different spanning subgraphs of G, in order to construct cochain groups. We also construct a degree preserving differential. When taking the graded Euler characteristic of the corresponding cohomology groups yielded by the cochain complex; we recover our version of the Tutte polynomial associated to G.; In this work we also study properties of the cohomology groups that we constructed and that we refer as Tutte homology. We present exact sequences associated to the deletion and contraction operations in graphs. We describe the effect on the cohomology groups when adding a pendant edge. In this case we are able to describe the generators of the Tutte Homology.; We determine a functorial property that assigns homomorphisms between homology groups to each inclusion map of graphs. Finally, we determine some connections between the two versions of the chromatic homology defined by Helme-Guizon-Rong and Stosic. |
