Long Exact Sequence of Khovanov Homology and Torsion

Khovanov homology (KH) is currently a very active research area in knot theory, in part, arguably, because of its close relation with the bracket polynomial and the Jones polynomial. This lecture offers an exposition of two concepts closely associated with Khovanov homology. First, we examine the fact that the Kauffman skein relation, which allows to define the Kauffman bracket polynomial up to normalization of the unknot, can be categorified by a long exact sequence. Second, we present one of the most elusive aspects of the theory, torsion in Khovanov homology. Aiming for a chronological showcase of the discoveries related to the torsion groups, we mention them in the order they were discovered. Notice that only at 2017 and 2019, infinite families of links with specific torsion orders were found, giving rise to a new wave of investigations.