Extreme and almost-extreme Khovanov spectra

At the end of the past century, Mikhail Khovanov introduced the first homological invariant, now known as Khovanov homology, as a categorification of Jones polynomial. It is a bigraded homology supported in homological and quantum gradings. Given a link diagram, we refer to the maximal quantum grading such that the associated Khovanov complex is non-trivial as extreme grading.
In this talk we discuss various properties of extreme and almost-extreme Khovanov homology and their geometric realizations. We will also study the relations of these models with Khovanov spectra, introduced by Robert Lipshitz and Sucharit Sarkar as a spatial refinement of Khovanov homology.