In this notebook we demonstrate the existence of a hyperbolic, fibered, SQP knot with non-vanishing first Khovanov homology.

In [1]:
import snappy
In [2]:
word=[3, 2, 2, 1, -2, 3, 2, 2, 1, 3, 2, 2, 1,1, 1, -2, -2, -2, -2]
In [3]:
D=snappy.Link(braid_closure=word)
In [4]:
D.exterior().verify_hyperbolicity()[0]
Out[4]:
True

This shows that the knot is hyperbolic.

In [5]:
D.knot_floer_homology()
Out[5]:
{'L_space_knot': False,
 'epsilon': 1,
 'fibered': True,
 'modulus': 2,
 'nu': 3,
 'ranks': {(-3, -6): 1,
  (-2, -5): 1,
  (-2, -4): 1,
  (-2, -3): 1,
  (-1, -3): 3,
  (-1, -2): 4,
  (0, -2): 5,
  (0, -1): 6,
  (1, -1): 3,
  (1, 0): 4,
  (2, -1): 1,
  (2, 0): 1,
  (2, 1): 1,
  (3, 0): 1},
 'seifert_genus': 3,
 'tau': 3,
 'total_rank': 33}

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.