In this notebook, we verify that our examples of Legendrian knots with the same Stein trace and arbitrarily negative tb from the contact (*n) from Theorem 1.2.(ii) move are really smoothly non-isotopic.

For that we load the surgery description of the L'_m and compare their volumes for short slopes and use the limit formula from Neumann-Zagier to deduce the result for the infinite family.

The knots L_m are smoothly all isotopic to L_0=K19n4 which has volume 7.79830023217? as computed in the notebook about the anulus twist.

In [1]:
import snappy
In [2]:
L=snappy.Manifold('surgery_description_contact_n_move.lnk')
In [3]:
def better_volume(M,index=100):
    '''Computes the verified volume. Returns 'unclear' if SnapPy could not do it.'''
    count=0
    while count<index:
        try:
            return M.volume(verified=True)
        except:
            M.randomize()
            count=count+1
    return 'unclear'
In [4]:
for n in range(1,100):
    L.dehn_fill([(1,n),(0,0)])
    print(n,L,better_volume(L))

1 unnamed link(1,1)(0,0) 14.0182393480?
2 unnamed link(1,2)(0,0) 15.301480974?
3 unnamed link(1,3)(0,0) 15.8026376515?
4 unnamed link(1,4)(0,0) 16.037542633?
5 unnamed link(1,5)(0,0) 16.1643932322?
6 unnamed link(1,6)(0,0) 16.2401827180?
7 unnamed link(1,7)(0,0) 16.2889341652?
8 unnamed link(1,8)(0,0) 16.322091751?
9 unnamed link(1,9)(0,0) 16.3456446593?
10 unnamed link(1,10)(0,0) 16.3629664020?
11 unnamed link(1,11)(0,0) 16.376072222?
12 unnamed link(1,12)(0,0) 16.386225136?
13 unnamed link(1,13)(0,0) 16.394248901?
14 unnamed link(1,14)(0,0) 16.400699149?
15 unnamed link(1,15)(0,0) 16.405961585?
16 unnamed link(1,16)(0,0) 16.410310681?
17 unnamed link(1,17)(0,0) 16.413946037?
18 unnamed link(1,18)(0,0) 16.417015598?
19 unnamed link(1,19)(0,0) 16.419630895?
20 unnamed link(1,20)(0,0) 16.421877287?
21 unnamed link(1,21)(0,0) 16.423820997?
22 unnamed link(1,22)(0,0) 16.4255140339?
23 unnamed link(1,23)(0,0) 16.4269977027?
24 unnamed link(1,24)(0,0) 16.428305142?
25 unnamed link(1,25)(0,0) 16.4294631845?
26 unnamed link(1,26)(0,0) 16.430493749?
27 unnamed link(1,27)(0,0) 16.431414874?
28 unnamed link(1,28)(0,0) 16.432241522?
29 unnamed link(1,29)(0,0) 16.432986180?
30 unnamed link(1,30)(0,0) 16.433659341?
31 unnamed link(1,31)(0,0) 16.434269871?
32 unnamed link(1,32)(0,0) 16.434825304?
33 unnamed link(1,33)(0,0) 16.435332074?
34 unnamed link(1,34)(0,0) 16.435795702?
35 unnamed link(1,35)(0,0) 16.436220951?
36 unnamed link(1,36)(0,0) 16.436611940?
37 unnamed link(1,37)(0,0) 16.436972253?
38 unnamed link(1,38)(0,0) 16.4373050165?
39 unnamed link(1,39)(0,0) 16.437612969?
40 unnamed link(1,40)(0,0) 16.437898517?
41 unnamed link(1,41)(0,0) 16.438163781?
42 unnamed link(1,42)(0,0) 16.438410638?
43 unnamed link(1,43)(0,0) 16.438640753?
44 unnamed link(1,44)(0,0) 16.438855605?
45 unnamed link(1,45)(0,0) 16.439056515?
46 unnamed link(1,46)(0,0) 16.439244663?
47 unnamed link(1,47)(0,0) 16.439421108?
48 unnamed link(1,48)(0,0) 16.439586801?
49 unnamed link(1,49)(0,0) 16.439742597?
50 unnamed link(1,50)(0,0) 16.439889270?
51 unnamed link(1,51)(0,0) 16.440027517?
52 unnamed link(1,52)(0,0) 16.440157973?
53 unnamed link(1,53)(0,0) 16.440281211?
54 unnamed link(1,54)(0,0) 16.440397755?
55 unnamed link(1,55)(0,0) 16.440508080?
56 unnamed link(1,56)(0,0) 16.4406126214?
57 unnamed link(1,57)(0,0) 16.440711777?
58 unnamed link(1,58)(0,0) 16.440805909?
59 unnamed link(1,59)(0,0) 16.440895353?
60 unnamed link(1,60)(0,0) 16.440980413?
61 unnamed link(1,61)(0,0) 16.441061373?
62 unnamed link(1,62)(0,0) 16.441138491?
63 unnamed link(1,63)(0,0) 16.441212006?
64 unnamed link(1,64)(0,0) 16.441282140?
65 unnamed link(1,65)(0,0) 16.441349097?
66 unnamed link(1,66)(0,0) 16.441413066?
67 unnamed link(1,67)(0,0) 16.441474222?
68 unnamed link(1,68)(0,0) 16.441532727?
69 unnamed link(1,69)(0,0) 16.441588733?
70 unnamed link(1,70)(0,0) 16.441642379?
71 unnamed link(1,71)(0,0) 16.441693797?
72 unnamed link(1,72)(0,0) 16.441743109?
73 unnamed link(1,73)(0,0) 16.441790427?
74 unnamed link(1,74)(0,0) 16.441835858?
75 unnamed link(1,75)(0,0) 16.441879501?
76 unnamed link(1,76)(0,0) 16.441921448?
77 unnamed link(1,77)(0,0) 16.441961787?
78 unnamed link(1,78)(0,0) 16.442000597?
79 unnamed link(1,79)(0,0) 16.442037957?
80 unnamed link(1,80)(0,0) 16.442073936?
81 unnamed link(1,81)(0,0) 16.442108603?
82 unnamed link(1,82)(0,0) 16.442142020?
83 unnamed link(1,83)(0,0) 16.442174246?
84 unnamed link(1,84)(0,0) 16.442205338?
85 unnamed link(1,85)(0,0) 16.442235348?
86 unnamed link(1,86)(0,0) 16.4422643258?
87 unnamed link(1,87)(0,0) 16.442292319?
88 unnamed link(1,88)(0,0) 16.442319370?
89 unnamed link(1,89)(0,0) 16.442345521?
90 unnamed link(1,90)(0,0) 16.442370813?
91 unnamed link(1,91)(0,0) 16.442395282?
92 unnamed link(1,92)(0,0) 16.442418963?
93 unnamed link(1,93)(0,0) 16.442441890?
94 unnamed link(1,94)(0,0) 16.442464095?
95 unnamed link(1,95)(0,0) 16.442485607?
96 unnamed link(1,96)(0,0) 16.442506456?
97 unnamed link(1,97)(0,0) 16.442526668?
98 unnamed link(1,98)(0,0) 16.442546268?
99 unnamed link(1,99)(0,0) 16.442565282?

This proves the claimed result and finishes the proof of Theorem 1.2.(ii) in the case of negative tb.