12n
0020
(K12n
0020
)
A knot diagram
1
Linearized knot diagam
3 5 6 7 2 10 5 12 6 9 8 11
Solving Sequence
6,10 2,7
5 3 8 1 4 9 11 12
c
6
c
5
c
2
c
7
c
1
c
4
c
9
c
10
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h6.15417 × 10
24
u
34
+ 4.90608 × 10
25
u
33
+ ··· + 1.32838 × 10
26
b − 6.91943 × 10
25
,
1.02387 × 10
26
u
34
+ 2.68565 × 10
26
u
33
+ ··· + 1.32838 × 10
26
a − 3.02927 × 10
26
, u
35
+ 3u
34
+ ··· − 2u − 1i
I
u
2
= h4u
5
a + 5u
4
a + 4u
5
− 7u
3
a + 5u
4
− 14u
2
a − 7u
3
+ 5au − 14u
2
+ 17b + a + 5u + 1,
− u
5
a + 2u
3
a − 2u
4
+ u
2
a − u
3
+ a
2
− 2au + 2u
2
+ 2u, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 47 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I.
I
u
1
= h6.15 × 10
24
u
34
+ 4.91 × 10
25
u
33
+ · · · + 1.33 × 10
26
b − 6.92 × 10
25
, 1.02 ×
10
26
u
34
+2.69×10
26
u
33
+· · ·+1.33×10
26
a−3.03×10
26
, u
35
+3u
34
+· · ·−2u−1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
2
=
−0.770764u
34
− 2.02174u
33
+ ··· − 4.41998u + 2.28042
−0.0463282u
34
− 0.369327u
33
+ ··· + 0.317345u + 0.520890
a
7
=
1
u
2
a
5
=
1.26890u
34
+ 2.84616u
33
+ ··· − 4.14141u + 0.782118
−0.0631189u
34
− 0.437170u
33
+ ··· + 0.413577u − 0.482385
a
3
=
1.55412u
34
+ 3.44078u
33
+ ··· − 4.40019u − 0.146776
−0.0448946u
34
− 0.427369u
33
+ ··· + 0.806983u − 0.450739
a
8
=
0.690375u
34
+ 2.08177u
33
+ ··· − 1.15296u − 2.07734
0.00686562u
34
+ 0.115030u
33
+ ··· − 0.332383u − 0.172479
a
1
=
0.700744u
34
+ 1.98651u
33
+ ··· − 0.108902u − 1.89421
0.0103696u
34
− 0.0952592u
33
+ ··· + 1.04406u + 0.183128
a
4
=
1.59901u
34
+ 3.86815u
33
+ ··· − 5.20717u + 0.303962
−0.0448946u
34
− 0.427369u
33
+ ··· + 0.806983u − 0.450739
a
9
=
u
u
a
11
=
−u
3
−u
3
+ u
a
12
=
0.743963u
34
+ 2.07828u
33
+ ··· − 0.147237u − 1.89598
0.0883859u
34
+ 0.102670u
33
+ ··· + 1.52166u + 0.0975759
(ii) Obstruction class = −1
(iii) Cusp Shapes =
138501011022536071388354471
22139732193931699544465805
u
34
+
368226358523013848196048853
22139732193931699544465805
u
33
+
··· −
423513858408885950087940661
44279464387863399088931610
u −
130309672706343830288860133
14759821462621133029643870
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
35
+ 5u
34
+ ··· + 6u − 1
c
2
, c
5
u
35
+ 7u
34
+ ··· − 6u − 1
c
3
u
35
− 7u
34
+ ··· − 25346u − 337
c
4
, c
7
u
35
+ 3u
34
+ ··· + 16384u + 4096
c
6
, c
9
u
35
+ 3u
34
+ ··· − 2u − 1
c
8
, c
11
u
35
+ 3u
34
+ ··· + 2u − 1
c
10
u
35
+ 3u
34
+ ··· − 2u + 1
c
12
u
35
− 23u
34
+ ··· − 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
35
+ 57y
34
+ ··· + 6y −1
c
2
, c
5
y
35
+ 5y
34
+ ··· + 6y −1
c
3
y
35
+ 109y
34
+ ··· + 279652022y −113569
c
4
, c
7
y
35
− 65y
34
+ ··· − 83886080y −16777216
c
6
, c
9
y
35
− 3y
34
+ ··· − 2y −1
c
8
, c
11
y
35
− 23y
34
+ ··· − 2y −1
c
10
y
35
+ 61y
34
+ ··· + 6y −1
c
12
y
35
− 19y
34
+ ··· + 182y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.991966 + 0.091776I
a = 0.72692 + 1.45829I
b = −0.068558 + 0.738981I
−2.80047 + 0.03393I −4.96122 − 0.73381I
u = −0.991966 − 0.091776I
a = 0.72692 − 1.45829I
b = −0.068558 − 0.738981I
−2.80047 − 0.03393I −4.96122 + 0.73381I
u = 0.909482 + 0.380777I
a = 0.29144 − 1.73248I
b = 0.014187 − 1.000160I
−1.72254 − 4.24984I −2.18876 + 7.04122I
u = 0.909482 − 0.380777I
a = 0.29144 + 1.73248I
b = 0.014187 + 1.000160I
−1.72254 + 4.24984I −2.18876 − 7.04122I
u = 0.576907 + 0.754246I
a = 0.591225 + 0.311198I
b = −0.377861 − 0.131381I
2.25585 + 1.15466I 4.96533 − 0.29519I
u = 0.576907 − 0.754246I
a = 0.591225 − 0.311198I
b = −0.377861 + 0.131381I
2.25585 − 1.15466I 4.96533 + 0.29519I
u = −1.010040 + 0.446446I
a = 0.650598 − 0.883449I
b = −0.388281 − 0.417527I
−1.67432 + 1.71265I −0.948963 + 0.233573I
u = −1.010040 − 0.446446I
a = 0.650598 + 0.883449I
b = −0.388281 + 0.417527I
−1.67432 − 1.71265I −0.948963 − 0.233573I
u = −0.438128 + 0.690005I
a = −0.45748 + 1.41127I
b = 0.70266 + 1.26431I
2.97461 + 5.04238I 7.52995 − 7.92929I
u = −0.438128 − 0.690005I
a = −0.45748 − 1.41127I
b = 0.70266 − 1.26431I
2.97461 − 5.04238I 7.52995 + 7.92929I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.125324 + 0.796091I
a = 0.172538 − 0.306733I
b = 1.204060 − 0.564978I
5.45936 − 1.99795I 12.38801 + 3.24689I
u = 0.125324 − 0.796091I
a = 0.172538 + 0.306733I
b = 1.204060 + 0.564978I
5.45936 + 1.99795I 12.38801 − 3.24689I
u = 1.114360 + 0.664977I
a = 0.458155 + 0.916311I
b = −0.612297 + 0.465157I
0.69794 − 6.72088I 4.01525 + 3.80549I
u = 1.114360 − 0.664977I
a = 0.458155 − 0.916311I
b = −0.612297 − 0.465157I
0.69794 + 6.72088I 4.01525 − 3.80549I
u = 0.675945
a = 2.49935
b = 0.470215
2.49299 1.75040
u = −0.019593 + 0.666979I
a = 0.386507 − 0.586859I
b = 0.438826 + 0.519059I
0.93795 + 1.36112I 3.65858 − 4.50590I
u = −0.019593 − 0.666979I
a = 0.386507 + 0.586859I
b = 0.438826 − 0.519059I
0.93795 − 1.36112I 3.65858 + 4.50590I
u = −0.000172 + 0.620501I
a = 0.266778 − 1.045710I
b = 0.458461 + 0.655103I
0.93048 + 1.37281I 3.33756 − 4.46340I
u = −0.000172 − 0.620501I
a = 0.266778 + 1.045710I
b = 0.458461 − 0.655103I
0.93048 − 1.37281I 3.33756 + 4.46340I
u = 0.428622 + 0.434204I
a = −1.22891 − 2.28547I
b = 0.489459 − 1.005660I
−0.28045 − 2.82979I 1.83395 + 3.29320I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.428622 − 0.434204I
a = −1.22891 + 2.28547I
b = 0.489459 + 1.005660I
−0.28045 + 2.82979I 1.83395 − 3.29320I
u = −0.98049 + 1.10389I
a = −0.491542 + 0.320489I
b = −1.14756 + 0.89148I
16.2670 + 5.6984I 6.10259 − 2.75484I
u = −0.98049 − 1.10389I
a = −0.491542 − 0.320489I
b = −1.14756 − 0.89148I
16.2670 − 5.6984I 6.10259 + 2.75484I
u = 0.99399 + 1.11491I
a = −0.412944 − 0.428999I
b = −1.046520 − 0.943327I
11.73730 − 0.22593I 0
u = 0.99399 − 1.11491I
a = −0.412944 + 0.428999I
b = −1.046520 + 0.943327I
11.73730 + 0.22593I 0
u = −1.10793 + 1.00267I
a = 0.35872 − 1.51332I
b = −1.04229 − 1.00409I
15.8086 + 2.0235I 5.61504 + 0.I
u = −1.10793 − 1.00267I
a = 0.35872 + 1.51332I
b = −1.04229 + 1.00409I
15.8086 − 2.0235I 5.61504 + 0.I
u = −1.08711 + 1.02841I
a = 0.53533 − 1.58797I
b = −0.94816 − 1.13708I
15.3839 + 13.3033I 4.97161 − 6.77945I
u = −1.08711 − 1.02841I
a = 0.53533 + 1.58797I
b = −0.94816 + 1.13708I
15.3839 − 13.3033I 4.97161 + 6.77945I
u = −1.01043 + 1.10512I
a = −0.436340 + 0.546687I
b = −1.01525 + 1.03906I
15.6838 − 5.5084I 5.51943 + 2.85249I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.01043 − 1.10512I
a = −0.436340 − 0.546687I
b = −1.01525 − 1.03906I
15.6838 + 5.5084I 5.51943 − 2.85249I
u = 1.10529 + 1.02256I
a = 0.46909 + 1.51898I
b = −0.96358 + 1.05901I
11.33260 − 7.58418I 0. + 4.10781I
u = 1.10529 − 1.02256I
a = 0.46909 − 1.51898I
b = −0.96358 − 1.05901I
11.33260 + 7.58418I 0. − 4.10781I
u = −0.446086 + 0.207143I
a = 5.87024 − 1.44569I
b = 0.567586 − 0.882988I
1.99036 − 2.28427I −9.4417 − 11.9389I
u = −0.446086 − 0.207143I
a = 5.87024 + 1.44569I
b = 0.567586 + 0.882988I
1.99036 + 2.28427I −9.4417 + 11.9389I
8
II.
I
u
2
= h4u
5
a+4u
5
+· · ·+a + 1, −u
5
a−2u
4
+· · ·+a
2
+2u, u
6
+u
5
−u
4
−2u
3
+u+1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
2
=
a
−0.235294au
5
− 0.235294u
5
+ ··· − 0.0588235a − 0.0588235
a
7
=
1
u
2
a
5
=
0.235294au
5
− 0.764706u
5
+ ··· + 1.05882a + 0.0588235
−0.235294au
5
− 0.235294u
5
+ ··· − 0.0588235a − 1.05882
a
3
=
−u
5
+ 2u
3
+ u
2
+ a − 2u − 1
−0.235294au
5
− 0.235294u
5
+ ··· − 0.0588235a − 1.05882
a
8
=
1
u
2
a
1
=
−1
0
a
4
=
0.235294au
5
− 0.764706u
5
+ ··· + 1.05882a + 0.0588235
−0.235294au
5
− 0.235294u
5
+ ··· − 0.0588235a − 1.05882
a
9
=
u
u
a
11
=
−u
3
−u
3
+ u
a
12
=
u
5
− 2u
3
+ u
u
5
+ u
4
− 2u
3
− u
2
+ u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
33
17
u
5
a +
3
17
u
4
a +
33
17
u
5
−
62
17
u
3
a +
88
17
u
4
−
56
17
u
2
a −
62
17
u
3
+
54
17
au −
124
17
u
2
+
4
17
a −
14
17
u +
106
17
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
6
c
2
(u
2
+ u + 1)
6
c
4
, c
7
u
12
c
6
, c
11
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
c
8
, c
9
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
c
10
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
2
c
12
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
6
c
4
, c
7
y
12
c
6
, c
8
, c
9
c
11
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
c
10
, c
12
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.002190 + 0.295542I
a = 0.315127 + 1.283850I
b = 0.500000 + 0.866025I
−1.89061 + 1.10558I 0.30406 − 2.63469I
u = 1.002190 + 0.295542I
a = −0.54572 − 1.78086I
b = 0.500000 − 0.866025I
−1.89061 − 2.95419I −2.90246 + 4.54482I
u = 1.002190 − 0.295542I
a = 0.315127 − 1.283850I
b = 0.500000 − 0.866025I
−1.89061 − 1.10558I 0.30406 + 2.63469I
u = 1.002190 − 0.295542I
a = −0.54572 + 1.78086I
b = 0.500000 + 0.866025I
−1.89061 + 2.95419I −2.90246 − 4.54482I
u = −0.428243 + 0.664531I
a = 0.431357 + 0.434984I
b = 0.500000 − 0.866025I
1.89061 − 2.95419I 2.82220 + 4.67955I
u = −0.428243 + 0.664531I
a = −2.09239 + 1.02210I
b = 0.500000 + 0.866025I
1.89061 + 1.10558I 6.66783 − 4.72351I
u = −0.428243 − 0.664531I
a = 0.431357 − 0.434984I
b = 0.500000 + 0.866025I
1.89061 + 2.95419I 2.82220 − 4.67955I
u = −0.428243 − 0.664531I
a = −2.09239 − 1.02210I
b = 0.500000 − 0.866025I
1.89061 − 1.10558I 6.66783 + 4.72351I
u = −1.073950 + 0.558752I
a = 0.179704 − 0.925804I
b = 0.500000 − 0.866025I
3.66314I 3.68173 − 3.33422I
u = −1.073950 + 0.558752I
a = −0.78808 + 1.48456I
b = 0.500000 + 0.866025I
7.72290I −0.57335 − 9.26831I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.073950 − 0.558752I
a = 0.179704 + 0.925804I
b = 0.500000 + 0.866025I
− 3.66314I 3.68173 + 3.33422I
u = −1.073950 − 0.558752I
a = −0.78808 − 1.48456I
b = 0.500000 − 0.866025I
− 7.72290I −0.57335 + 9.26831I
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
6
)(u
35
+ 5u
34
+ ··· + 6u − 1)
c
2
((u
2
+ u + 1)
6
)(u
35
+ 7u
34
+ ··· − 6u − 1)
c
3
((u
2
− u + 1)
6
)(u
35
− 7u
34
+ ··· − 25346u − 337)
c
4
, c
7
u
12
(u
35
+ 3u
34
+ ··· + 16384u + 4096)
c
5
((u
2
− u + 1)
6
)(u
35
+ 7u
34
+ ··· − 6u − 1)
c
6
((u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
)(u
35
+ 3u
34
+ ··· − 2u − 1)
c
8
((u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
)(u
35
+ 3u
34
+ ··· + 2u − 1)
c
9
((u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
)(u
35
+ 3u
34
+ ··· − 2u − 1)
c
10
((u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
2
)(u
35
+ 3u
34
+ ··· − 2u + 1)
c
11
((u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
)(u
35
+ 3u
34
+ ··· + 2u − 1)
c
12
((u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
)(u
35
− 23u
34
+ ··· − 2u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
6
)(y
35
+ 57y
34
+ ··· + 6y −1)
c
2
, c
5
((y
2
+ y + 1)
6
)(y
35
+ 5y
34
+ ··· + 6y −1)
c
3
((y
2
+ y + 1)
6
)(y
35
+ 109y
34
+ ··· + 2.79652 × 10
8
y −113569)
c
4
, c
7
y
12
(y
35
− 65y
34
+ ··· − 8.38861 × 10
7
y −1.67772 × 10
7
)
c
6
, c
9
((y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
)(y
35
− 3y
34
+ ··· − 2y −1)
c
8
, c
11
((y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
)(y
35
− 23y
34
+ ··· − 2y −1)
c
10
((y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
)(y
35
+ 61y
34
+ ··· + 6y −1)
c
12
((y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
)(y
35
− 19y
34
+ ··· + 182y −1)
15
