In this notebook we demonstrate the existence of a hyperbolic, fibered, SQP knot with non-vanishing first Khovanov homology.
import snappy
word=[3, 2, 2, 1, -2, 3, 2, 2, 1, 3, 2, 2, 1,1, 1, -2, -2, -2, -2]
D=snappy.Link(braid_closure=word)
D.exterior().verify_hyperbolicity()[0]
This shows that the knot is hyperbolic.
D.knot_floer_homology()
Since it is fibered and g3=tau, it is SQP. Then we can load that knot in KnotJob, compute its Khovanov homology and observe that its first Khovanov homology is non-vanishing.