12n
0081
(K12n
0081
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 9 3 10 12 6 1 8 11
Solving Sequence
5,9 3,6
7 10 2 1 11 4 12 8
c
5
c
6
c
9
c
2
c
1
c
10
c
4
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h5.53432 × 10
85
u
57
− 8.84907 × 10
85
u
56
+ ··· + 1.15940 × 10
86
b − 1.39649 × 10
85
,
2.61940 × 10
85
u
57
− 5.40341 × 10
85
u
56
+ ··· + 2.31879 × 10
85
a − 2.38511 × 10
85
, u
58
− 2u
57
+ ··· − u + 1i
I
u
2
= hb + 1, 2u
7
+ 3u
6
− 5u
5
− 7u
4
+ 4u
3
+ 3u
2
+ a + 4, u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 66 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
= h5.53 × 10
85
u
57
− 8.85 × 10
85
u
56
+ · · · + 1.16 × 10
86
b − 1.40 × 10
85
, 2.62 ×
10
85
u
57
−5.40×10
85
u
56
+· · ·+2.32×10
85
a−2.39×10
85
, u
58
−2u
57
+· · ·−u+1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
3
=
−1.12964u
57
+ 2.33027u
56
+ ··· − 0.0189625u + 1.02860
−0.477346u
57
+ 0.763249u
56
+ ··· + 1.48654u + 0.120450
a
6
=
1
u
2
a
7
=
−0.742987u
57
+ 1.16928u
56
+ ··· + 2.16233u + 0.683441
−0.147869u
57
+ 0.246417u
56
+ ··· + 0.573664u + 0.388653
a
10
=
−u
−u
3
+ u
a
2
=
−1.60699u
57
+ 3.09352u
56
+ ··· + 1.46757u + 1.14905
−0.477346u
57
+ 0.763249u
56
+ ··· + 1.48654u + 0.120450
a
1
=
−0.768033u
57
+ 1.20204u
56
+ ··· + 2.30970u + 0.755400
0.0250459u
57
− 0.0327586u
56
+ ··· − 0.147371u − 0.0719590
a
11
=
0.238196u
57
− 0.406882u
56
+ ··· − 2.56618u − 0.552552
−0.0179618u
57
+ 0.0170715u
56
+ ··· + 0.925015u − 0.0202253
a
4
=
−1.20110u
57
+ 2.45056u
56
+ ··· + 0.266941u + 1.22004
−0.484972u
57
+ 0.772733u
56
+ ··· + 1.53536u + 0.143087
a
12
=
−0.842522u
57
+ 1.44178u
56
+ ··· + 3.31034u + 1.38391
0.240620u
57
− 0.365565u
56
+ ··· − 0.386959u − 0.465919
a
8
=
−0.606276u
57
+ 0.938630u
56
+ ··· + 1.72832u + 0.349414
−0.275800u
57
+ 0.459793u
56
+ ··· + 0.913732u + 0.679907
(ii) Obstruction class = −1
(iii) Cusp Shapes = −1.20777u
57
+ 0.883719u
56
+ ··· − 4.35534u − 9.32260
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
58
+ 19u
57
+ ··· + 1227u + 1
c
2
, c
4
u
58
− 9u
57
+ ··· − 43u + 1
c
3
, c
6
u
58
+ 7u
57
+ ··· + 2688u + 256
c
5
, c
9
u
58
+ 2u
57
+ ··· + u + 1
c
7
u
58
− 2u
57
+ ··· + 42759u + 8017
c
8
, c
11
u
58
+ 2u
57
+ ··· + 7u + 1
c
10
, c
12
u
58
+ 18u
57
+ ··· + 11u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
58
+ 49y
57
+ ··· − 1420135y + 1
c
2
, c
4
y
58
− 19y
57
+ ··· − 1227y + 1
c
3
, c
6
y
58
− 51y
57
+ ··· − 3719168y + 65536
c
5
, c
9
y
58
− 14y
57
+ ··· − 11y + 1
c
7
y
58
+ 22y
57
+ ··· + 71584681y + 64272289
c
8
, c
11
y
58
− 18y
57
+ ··· − 11y + 1
c
10
, c
12
y
58
+ 46y
57
+ ··· − 251y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.757964 + 0.476676I
a = 0.211091 − 1.285110I
b = −0.733524 + 1.001690I
1.39650 + 7.38606I −8.93309 − 9.73199I
u = −0.757964 − 0.476676I
a = 0.211091 + 1.285110I
b = −0.733524 − 1.001690I
1.39650 − 7.38606I −8.93309 + 9.73199I
u = 0.733536 + 0.503634I
a = 0.207333 + 1.258410I
b = −0.622314 − 0.960850I
1.93231 − 1.77262I −7.23531 + 4.32887I
u = 0.733536 − 0.503634I
a = 0.207333 − 1.258410I
b = −0.622314 + 0.960850I
1.93231 + 1.77262I −7.23531 − 4.32887I
u = 0.150361 + 0.862598I
a = 0.283573 + 0.156210I
b = 0.296125 − 0.128802I
1.86705 − 2.42873I −2.94093 + 3.38399I
u = 0.150361 − 0.862598I
a = 0.283573 − 0.156210I
b = 0.296125 + 0.128802I
1.86705 + 2.42873I −2.94093 − 3.38399I
u = 0.729010 + 0.900450I
a = −0.278287 + 0.714115I
b = 0.512624 − 0.941655I
1.79327 − 3.50993I −8.00000 + 0.I
u = 0.729010 − 0.900450I
a = −0.278287 − 0.714115I
b = 0.512624 + 0.941655I
1.79327 + 3.50993I −8.00000 + 0.I
u = −0.707674 + 0.373154I
a = 0.190149 − 1.303650I
b = −0.946128 + 0.714336I
−3.61079 + 2.96792I −16.6955 − 7.6738I
u = −0.707674 − 0.373154I
a = 0.190149 + 1.303650I
b = −0.946128 − 0.714336I
−3.61079 − 2.96792I −16.6955 + 7.6738I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.828609 + 0.884235I
a = −0.466481 + 0.852083I
b = 0.613518 − 1.215300I
8.38484 − 8.22953I 0
u = 0.828609 − 0.884235I
a = −0.466481 − 0.852083I
b = 0.613518 + 1.215300I
8.38484 + 8.22953I 0
u = −0.827954 + 0.902716I
a = −0.485901 − 0.807428I
b = 0.663928 + 1.180430I
9.06399 + 2.20407I 0
u = −0.827954 − 0.902716I
a = −0.485901 + 0.807428I
b = 0.663928 − 1.180430I
9.06399 − 2.20407I 0
u = 0.758547 + 0.131049I
a = −0.057075 + 0.644979I
b = −1.47907 − 0.32636I
−1.42978 − 4.33965I −13.8036 + 5.9098I
u = 0.758547 − 0.131049I
a = −0.057075 − 0.644979I
b = −1.47907 + 0.32636I
−1.42978 + 4.33965I −13.8036 − 5.9098I
u = −0.738708 + 0.184329I
a = −0.061286 − 0.888067I
b = −1.37022 + 0.42544I
−1.16921 − 0.90179I −13.10774 − 0.16863I
u = −0.738708 − 0.184329I
a = −0.061286 + 0.888067I
b = −1.37022 − 0.42544I
−1.16921 + 0.90179I −13.10774 + 0.16863I
u = −0.771402 + 0.979679I
a = −0.418138 − 0.593129I
b = 0.735262 + 0.914814I
4.73839 + 0.20490I 0
u = −0.771402 − 0.979679I
a = −0.418138 + 0.593129I
b = 0.735262 − 0.914814I
4.73839 − 0.20490I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.725261
a = −0.320581
b = −1.46548
−5.24110 −20.4300
u = 1.010850 + 0.787058I
a = 0.82799 − 1.23601I
b = 0.817179 + 0.885590I
7.78595 + 1.97085I 0
u = 1.010850 − 0.787058I
a = 0.82799 + 1.23601I
b = 0.817179 − 0.885590I
7.78595 − 1.97085I 0
u = 0.747268 + 1.058050I
a = −0.395621 + 0.438566I
b = 0.813908 − 0.746877I
1.08274 + 2.68379I 0
u = 0.747268 − 1.058050I
a = −0.395621 − 0.438566I
b = 0.813908 + 0.746877I
1.08274 − 2.68379I 0
u = 0.562499 + 0.418736I
a = 0.47527 + 1.47284I
b = −0.667885 − 0.412194I
−0.84022 − 1.37563I −6.44015 + 4.83399I
u = 0.562499 − 0.418736I
a = 0.47527 − 1.47284I
b = −0.667885 + 0.412194I
−0.84022 + 1.37563I −6.44015 − 4.83399I
u = −1.021400 + 0.806453I
a = 0.76138 + 1.27349I
b = 0.872213 − 0.895148I
8.43324 + 4.16970I 0
u = −1.021400 − 0.806453I
a = 0.76138 − 1.27349I
b = 0.872213 + 0.895148I
8.43324 − 4.16970I 0
u = 0.464370 + 0.505657I
a = 2.19485 + 0.67921I
b = −0.485517 + 0.294137I
2.55163 − 1.79957I −5.78089 + 4.77562I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.464370 − 0.505657I
a = 2.19485 − 0.67921I
b = −0.485517 − 0.294137I
2.55163 + 1.79957I −5.78089 − 4.77562I
u = 1.302260 + 0.215056I
a = 0.724570 − 0.182519I
b = 0.657572 + 0.133991I
−1.94270 − 1.43075I 0
u = 1.302260 − 0.215056I
a = 0.724570 + 0.182519I
b = 0.657572 − 0.133991I
−1.94270 + 1.43075I 0
u = −0.429870 + 0.520549I
a = 2.51794 − 0.86930I
b = −0.605155 − 0.307118I
2.25105 − 3.87043I −6.73442 + 0.34200I
u = −0.429870 − 0.520549I
a = 2.51794 + 0.86930I
b = −0.605155 + 0.307118I
2.25105 + 3.87043I −6.73442 − 0.34200I
u = −0.831079 + 1.050820I
a = −0.560219 − 0.451422I
b = 0.967229 + 0.864793I
8.14146 − 2.33310I 0
u = −0.831079 − 1.050820I
a = −0.560219 + 0.451422I
b = 0.967229 − 0.864793I
8.14146 + 2.33310I 0
u = 0.834599 + 1.071090I
a = −0.562906 + 0.405830I
b = 0.998315 − 0.821896I
7.22076 + 8.31025I 0
u = 0.834599 − 1.071090I
a = −0.562906 − 0.405830I
b = 0.998315 + 0.821896I
7.22076 − 8.31025I 0
u = 1.113430 + 0.804823I
a = 0.544448 − 1.077950I
b = 0.972357 + 0.706110I
0.57880 − 2.87211I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.113430 − 0.804823I
a = 0.544448 + 1.077950I
b = 0.972357 − 0.706110I
0.57880 + 2.87211I 0
u = −1.089310 + 0.851827I
a = 0.480537 + 1.230050I
b = 1.055730 − 0.794375I
3.73913 + 6.53652I 0
u = −1.089310 − 0.851827I
a = 0.480537 − 1.230050I
b = 1.055730 + 0.794375I
3.73913 − 6.53652I 0
u = −1.083120 + 0.904822I
a = 0.329084 + 1.363660I
b = 1.19685 − 0.84814I
7.31786 + 9.44466I 0
u = −1.083120 − 0.904822I
a = 0.329084 − 1.363660I
b = 1.19685 + 0.84814I
7.31786 − 9.44466I 0
u = 1.08928 + 0.91350I
a = 0.284481 − 1.359270I
b = 1.22723 + 0.83313I
6.3813 − 15.5051I 0
u = 1.08928 − 0.91350I
a = 0.284481 + 1.359270I
b = 1.22723 − 0.83313I
6.3813 + 15.5051I 0
u = 1.11634 + 0.88115I
a = 0.344161 − 1.208590I
b = 1.144810 + 0.743437I
−0.06948 − 9.72018I 0
u = 1.11634 − 0.88115I
a = 0.344161 + 1.208590I
b = 1.144810 − 0.743437I
−0.06948 + 9.72018I 0
u = −1.43228 + 0.28779I
a = 0.602581 + 0.217299I
b = 0.746089 − 0.144890I
−3.47621 + 6.76006I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.43228 − 0.28779I
a = 0.602581 − 0.217299I
b = 0.746089 + 0.144890I
−3.47621 − 6.76006I 0
u = −1.47635
a = 0.603436
b = 0.732744
−7.57251 0
u = −0.310564 + 0.379876I
a = 2.83067 − 3.52170I
b = −0.888180 − 0.051391I
−2.59810 − 0.31577I −24.5055 − 6.4182I
u = −0.310564 − 0.379876I
a = 2.83067 + 3.52170I
b = −0.888180 + 0.051391I
−2.59810 + 0.31577I −24.5055 + 6.4182I
u = −0.485729
a = −2.58176
b = −1.10599
−2.22309 1.58660
u = −0.030516 + 0.465574I
a = 7.05708 − 0.52655I
b = −1.088060 − 0.022657I
0.94270 + 2.75058I 17.8156 − 8.6403I
u = −0.030516 − 0.465574I
a = 7.05708 + 0.52655I
b = −1.088060 + 0.022657I
0.94270 − 2.75058I 17.8156 + 8.6403I
u = 0.418573
a = 1.13637
b = 0.0289678
−0.881313 −11.5040
10
II. I
u
2
=
hb+1, 2u
7
+3u
6
−5u
5
−7u
4
+4u
3
+3u
2
+a+4, u
8
+u
7
−3u
6
−2u
5
+3u
4
+2u−1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
3
=
−2u
7
− 3u
6
+ 5u
5
+ 7u
4
− 4u
3
− 3u
2
− 4
−1
a
6
=
1
u
2
a
7
=
1
u
2
a
10
=
−u
−u
3
+ u
a
2
=
−2u
7
− 3u
6
+ 5u
5
+ 7u
4
− 4u
3
− 3u
2
− 5
−1
a
1
=
−1
0
a
11
=
u
3
− 2u
−u
3
+ u
a
4
=
−2u
7
− 3u
6
+ 5u
5
+ 7u
4
− 4u
3
− 3u
2
− 4
−1
a
12
=
−u
6
+ 3u
4
− 2u
2
− 1
u
6
− 2u
4
+ u
2
a
8
=
−u
2
+ 1
−u
4
+ 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u
7
− 16u
6
+ 18u
5
+ 36u
4
− 15u
3
− 13u
2
+ 4u − 37
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
6
u
8
c
4
(u + 1)
8
c
5
, c
7
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
8
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
9
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
10
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
11
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
12
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
6
y
8
c
5
, c
7
, c
9
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
8
, c
11
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
10
, c
12
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.180120 + 0.268597I
a = −0.615431 + 0.295452I
b = −1.00000
−2.68559 − 1.13123I −13.04860 − 0.79986I
u = 1.180120 − 0.268597I
a = −0.615431 − 0.295452I
b = −1.00000
−2.68559 + 1.13123I −13.04860 + 0.79986I
u = 0.108090 + 0.747508I
a = 1.68119 + 0.49658I
b = −1.00000
0.51448 − 2.57849I −11.13007 + 2.07507I
u = 0.108090 − 0.747508I
a = 1.68119 − 0.49658I
b = −1.00000
0.51448 + 2.57849I −11.13007 − 2.07507I
u = −1.37100
a = −0.532015
b = −1.00000
−8.14766 −21.6800
u = −1.334530 + 0.318930I
a = −0.473764 − 0.240160I
b = −1.00000
−4.02461 + 6.44354I −15.6905 − 2.6628I
u = −1.334530 − 0.318930I
a = −0.473764 + 0.240160I
b = −1.00000
−4.02461 − 6.44354I −15.6905 + 2.6628I
u = 0.463640
a = −4.65198
b = −1.00000
−2.48997 −37.5820
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
58
+ 19u
57
+ ··· + 1227u + 1)
c
2
((u − 1)
8
)(u
58
− 9u
57
+ ··· − 43u + 1)
c
3
, c
6
u
8
(u
58
+ 7u
57
+ ··· + 2688u + 256)
c
4
((u + 1)
8
)(u
58
− 9u
57
+ ··· − 43u + 1)
c
5
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
58
+ 2u
57
+ ··· + u + 1)
c
7
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
58
− 2u
57
+ ··· + 42759u + 8017)
c
8
(u
8
− u
7
+ ··· + 2u − 1)(u
58
+ 2u
57
+ ··· + 7u + 1)
c
9
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
58
+ 2u
57
+ ··· + u + 1)
c
10
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
58
+ 18u
57
+ ··· + 11u + 1)
c
11
(u
8
+ u
7
+ ··· − 2u − 1)(u
58
+ 2u
57
+ ··· + 7u + 1)
c
12
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
58
+ 18u
57
+ ··· + 11u + 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
58
+ 49y
57
+ ··· − 1420135y + 1)
c
2
, c
4
((y −1)
8
)(y
58
− 19y
57
+ ··· − 1227y + 1)
c
3
, c
6
y
8
(y
58
− 51y
57
+ ··· − 3719168y + 65536)
c
5
, c
9
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
58
− 14y
57
+ ··· − 11y + 1)
c
7
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
58
+ 22y
57
+ ··· + 71584681y + 64272289)
c
8
, c
11
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
58
− 18y
57
+ ··· − 11y + 1)
c
10
, c
12
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
58
+ 46y
57
+ ··· − 251y + 1)
16
