Khovanov homology is a link invariant introduced by Mikhail Khovanov as a categorification of Jones polynomial. It detects the unknot and provides geometric and topological information about knots (for example, it gives a lower bound for the 4-genus and detects fiberedness among positive links). Computing Jones polynomial (and therefore Khovanov homology) is an NP-hard problem. However, Morton and Short proved that, when fixing the number of strands of a braid, computing the Jones polynomial of its closure can be done in polynomial time with respect to the length of the braid. In this talk we present a conjecture on an equivalent statement for Khovanov homology, and introduce some results supporting it. On the way, we will work with (a simplified version of) Khovanov spectra, a link invariant introduced by Robert Lipshitz and Sucharit Sarkar as a refinement of Khovanov homology.
