Khovanov Homology: Unraveling Mysterious Algebraic Structures of Knots

 

Algebra plays an important role in low-dimensional topology and, in particular, one of its branches, knot theory. Most modern techniques to tell knots or manifolds apart are based on sophisticated algebraic constructions. One such construction which was introduced at the turn of the millennium and is still enjoying a surge of interest, is called categorification. Khovanov homology is one of the prime and most combinatorial examples of the categorification. This talk will start with a quick overview of the history of knot theory and a short introduction into knot invariants. I will then briefly discuss the Jones polynomial and explain how to generalize it into different versions of Khovanov homology. In the last 15 minutes, I will present new results related to certain homological operations that connect even and odd Khovanov homology theories. While the algebraic structure generated by these operations still remain a mystery, they appear to result in new knot invariants with unexpected properties.