Seminarium KMMF "Teoria Dwoistości"

We recall the definition of the Khovanov homology of links and mention its modification to odd Khovanov homology. We discuss the fact that in Khovanov homology of links, presence of ℤ_2-torsion is a very common phenomenon. We outline the several reasons for this observation (e.g., Hochschild homology of the algebra ℤ[x]/(x^2), crucial in Khovanov homology, has only ℤ_2 torsion). For other than ℤ_2 torsion only a finite number of examples of knots with ℤ_n-torsion were known, none for n>8. In this talk, we show that there are infinite families of links whose Khovanov homology contains ℤ_n-torsion for 2 < n < 9 and ℤ_(2^s)-torsion for s < 24. We also introduce 4-braid links with ℤ_3-torsion which are counterexamples to the PS braid conjecture. We mention that our construction also provides an infinite family of knots with ℤ_5-torsion in reduced Khovanov homology and ℤ_3-torsion in odd Khovanov homology. Most of our examples are twisted torus knots and links.