This thesis is in essence about the most fundamental problem of knot theory, that is, knot classification. We can imagine a knot as $S^1$ embedded in the three dimensional Euclidean space ${\mathbb R}^3$. More generally, we can talk about links. We can imagine these as finitely many knots, which are in some way interlaced. Given two arbitrary links, we can ask ourselves if it is possible to deform one into the other without cutting or tearing it anywhere in the process. We can often give an answer to this question with the use of link invariants. Link invariants are maps on the set of links, which map any two links that can be deformed to one another to the same image. Therefore, if a link invariant assigns to two links different images, we can conclude that it is in fact not possible to deform one to the other. Nowadays we know of many invariants. One of them is Khovanov homology, which is actually an invariant on the set of oriented links. Its construction relies heavily on tools from algebraic topology, and it turns out that there is a beautiful connection between this homology and another link invariant called Jones polynomial.
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