UC San Diego

In this dissertation we work with Khovanov homology and its variants. Khovanov homology is a "categorication" of the Jones polynomial. It consists of graded chain complexes which, up to chain homotopy, are link invariants, and whose graded Euler characteristic is equal to the Jones polynomial of the link. Rasmussen's invariant gives a bound on the smooth 4-ball genus of a knot. We construct bounds on Rasmussen's invariant which are easily computable from any diagram. Using this construction, one also obtains representatives for the homology classes of the Lee homology of the knot. These bounds are sharp precisely for homogeneous knots, a class of knots containing both alternating knots and positive knots. We prove that a class of braid-positive links, more general than torus links, are Khovanov thick. From this observation we get innite families of prime knots all of which are Khovanov thick. This is further evidence toward Khovanov's conjecture that all braid positive knots other than the T(2, 2k + 1) torus knots are thick. We provide a technique for creating cobordisms between knots which result in injections at the homological level. The technique is applied and studied in the case of ribbon knots