Khovanov homology of links and graphs

Abstract

In this thesis we work with Khovanov homology of links and its generalizations, as well as with the homology of graphs. Khovanov homology of links consists of graded chain complexes which are link invariants, up to chain homotopy, with graded Euler characteristic equal to the Jones polynomial of the link. Hence, it can be regarded as the "categorif\mbox{}ication" of the Jones polynomial. \indent We prove that the f\mbox{}irst homology group of positive braid knots is trivial. Futhermore, we prove that non-alternating torus knots are homologically thick. In addition, we show that we can decrease the number of full twists of torus knots without changing low-degree homology and consequently that there exists stable homology for torus knots. We also prove most of the above properties for Khovanov-Rozansky homology. \indent Concerning graph homology, we categorify the dichromatic (and consequently Tutte) polynomial for graphs, by categorifying an inf\mbox{}inite set of its one-variable specializations. We categorify explicitly the one-variable specialization that is an analog of the Jones polynomial of an alternating link corresponding to the initial graph. Also, we categorify explicitly the whole two-variable dichromatic polynomial of graphs by using Koszul complexes. \textbf{Key-words:} Khovanov homology, Jones polynomial, link, torus knot, graph, dichromatic polynomial