A FAST ALGORITHM FOR CALCULATING S-INVARIANTS

We use the divide-and-conquer and scanning algorithms for calculating Khovanov cohomology directly on the Lee- or Bar-Natan deformations of the Khovanov complex to give an alternative way to compute Rasmussen s-invariants of knots. By disregarding generators away from homological degree 0, we can considerably improve the efficiency of the algorithm. With a slight modification, we can also apply it to a refinement of Lipshitz–Sarkar.

Torsion calculations in Khovanov cohomology

We obtain information on torsion in Khovanov cohomology by performing calculations directly over [Formula: see text] for [Formula: see text] prime and [Formula: see text]. In particular, we get that the torus knots [Formula: see text] and [Formula: see text] contain torsion of order [Formula: see text] and [Formula: see text] in their Khovanov cohomology.

A geometric description of the extreme Khovanov cohomology

We prove that the potential extreme Khovanov cohomology of a link is the cohomology of the independence simplicial complex of its Lando graph. We also provide a family of knots having as many non-trivial extreme Khovanov cohomology modules as desired, that is, examples of H-thick knots that are as far from being H-thin as desired.