In this notebook we finish the proof of Theorem 1.3. We first show that the Legendrian knots L and L' are smoothly non-isotopic. For that we verify their hyperbolicity and check that their volumes do not agree. In a second step we show that the symmetry groups of their Legendrian surgeries both vanish and thus any diffeomorphism between them is smoothly isotopic to the contactomorphism we have constructed in Figure 23.

In [1]:
import snappy
In [2]:
KB=snappy.Manifold('KB.lnk')
KB.verify_hyperbolicity()[0]
Out[2]:
True
In [3]:
KG=snappy.Manifold('KG.lnk')
KG.verify_hyperbolicity()[0]
Out[3]:
True
In [4]:
KB.volume(verified=True)
Out[4]:
23.27073610?
In [5]:
KG.volume(verified=True)
Out[5]:
25.29931683?
In [6]:
KB.dehn_fill((0,1))
KB.symmetry_group()
Out[6]:
0
In [7]:
KG.dehn_fill((0,1))
KG.symmetry_group()
Out[7]:
0