12n
0104
(K12n
0104
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 10 3 11 12 6 1 8 9
Solving Sequence
5,10 3,6
7 2 1 11 4 9 12 8
c
5
c
6
c
2
c
1
c
10
c
4
c
9
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−3.59372 × 10
41
u
41
− 5.66142 × 10
41
u
40
+ ··· + 7.73697 × 10
41
b − 8.41893 × 10
39
,
1.32363 × 10
42
u
41
+ 2.48624 × 10
42
u
40
+ ··· + 7.73697 × 10
41
a + 6.53537 × 10
40
, u
42
+ 2u
41
+ ··· + u + 1i
I
u
2
= hb + 1, u
5
− 2u
4
+ 4u
3
− 5u
2
+ a + 4u − 3, u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 48 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
= h−3.59×10
41
u
41
−5.66×10
41
u
40
+· · ·+7.74×10
41
b−8.42×10
39
, 1.32×
10
42
u
41
+2.49×10
42
u
40
+· · ·+7.74×10
41
a+6.54×10
40
, u
42
+2u
41
+· · ·+u+1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
3
=
−1.71079u
41
− 3.21345u
40
+ ··· + 10.5415u − 0.0844693
0.464486u
41
+ 0.731736u
40
+ ··· − 1.23542u + 0.0108814
a
6
=
1
−u
2
a
7
=
1.20935u
41
+ 1.90913u
40
+ ··· − 3.32862u + 1.65557
−0.182245u
41
− 0.291497u
40
+ ··· + 0.535851u − 0.403264
a
2
=
−1.24630u
41
− 2.48172u
40
+ ··· + 9.30604u − 0.0735879
0.464486u
41
+ 0.731736u
40
+ ··· − 1.23542u + 0.0108814
a
1
=
1.24139u
41
+ 1.95441u
40
+ ··· − 3.49255u + 1.76187
−0.0320409u
41
− 0.0452803u
40
+ ··· + 0.163934u − 0.106300
a
11
=
0.722940u
41
+ 1.09186u
40
+ ··· − 1.22903u + 1.55759
−0.0729864u
41
− 0.0536384u
40
+ ··· + 1.18829u − 0.138312
a
4
=
−1.79877u
41
− 3.35612u
40
+ ··· + 10.8087u − 0.281705
0.474629u
41
+ 0.743231u
40
+ ··· − 1.29010u + 0.0441881
a
9
=
−u
u
3
+ u
a
12
=
0.986423u
41
+ 1.55821u
40
+ ··· − 2.67406u + 1.14887
0.258550u
41
+ 0.408387u
40
+ ··· − 0.795796u + 0.620421
a
8
=
−0.549074u
41
− 0.889586u
40
+ ··· + 0.0457032u − 1.27204
−0.183879u
41
− 0.191517u
40
+ ··· + 1.29243u − 0.478058
(ii) Obstruction class = −1
(iii) Cusp Shapes = −7.55767u
41
− 15.9158u
40
+ ··· + 73.6479u + 19.0106
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
42
+ 13u
41
+ ··· + 804u + 1
c
2
, c
4
u
42
− 7u
41
+ ··· − 32u + 1
c
3
, c
6
u
42
+ 5u
41
+ ··· − 256u + 64
c
5
, c
9
u
42
− 2u
41
+ ··· − u + 1
c
7
, c
8
, c
11
c
12
u
42
− 2u
41
+ ··· + u + 1
c
10
u
42
+ 14u
41
+ ··· + 5149u + 1583
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
42
+ 39y
41
+ ··· − 608644y + 1
c
2
, c
4
y
42
− 13y
41
+ ··· − 804y + 1
c
3
, c
6
y
42
− 39y
41
+ ··· − 253952y + 4096
c
5
, c
9
y
42
+ 10y
41
+ ··· − 7y + 1
c
7
, c
8
, c
11
c
12
y
42
− 50y
41
+ ··· − 7y + 1
c
10
y
42
− 26y
41
+ ··· − 65134235y + 2505889
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.01452
a = 0.159707
b = 0.426201
7.18041 14.5060
u = 0.501113 + 0.767004I
a = 0.191771 − 1.283310I
b = −0.669285 + 1.060730I
7.44472 + 4.92121I 8.52470 − 6.69865I
u = 0.501113 − 0.767004I
a = 0.191771 + 1.283310I
b = −0.669285 − 1.060730I
7.44472 − 4.92121I 8.52470 + 6.69865I
u = −0.453853 + 0.695369I
a = 0.247010 + 1.284860I
b = −0.719295 − 0.792687I
−0.28964 − 3.57536I 6.37219 + 9.69003I
u = −0.453853 − 0.695369I
a = 0.247010 − 1.284860I
b = −0.719295 + 0.792687I
−0.28964 + 3.57536I 6.37219 − 9.69003I
u = −0.161979 + 0.776395I
a = 0.051090 + 0.752460I
b = −1.48780 − 0.42283I
4.30541 − 1.84849I 3.64107 + 3.02333I
u = −0.161979 − 0.776395I
a = 0.051090 − 0.752460I
b = −1.48780 + 0.42283I
4.30541 + 1.84849I 3.64107 − 3.02333I
u = 0.912511 + 0.794551I
a = −0.419329 − 0.751716I
b = 0.631065 + 1.077450I
6.12642 + 3.75190I 10.63580 − 4.55895I
u = 0.912511 − 0.794551I
a = −0.419329 + 0.751716I
b = 0.631065 − 1.077450I
6.12642 − 3.75190I 10.63580 + 4.55895I
u = −0.891381 + 0.841839I
a = −0.506949 + 0.850686I
b = 0.658034 − 1.240570I
14.7689 − 5.7297I 11.79655 + 3.54668I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.891381 − 0.841839I
a = −0.506949 − 0.850686I
b = 0.658034 + 1.240570I
14.7689 + 5.7297I 11.79655 − 3.54668I
u = −0.985080 + 0.753794I
a = −0.386102 + 0.573806I
b = 0.715924 − 0.874389I
3.30789 − 0.29459I 6.00000 + 0.I
u = −0.985080 − 0.753794I
a = −0.386102 − 0.573806I
b = 0.715924 + 0.874389I
3.30789 + 0.29459I 6.00000 + 0.I
u = −0.803175 + 1.000510I
a = 0.83087 − 1.30330I
b = 0.831506 + 0.932621I
14.2431 − 0.5975I 11.31138 + 0.I
u = −0.803175 − 1.000510I
a = 0.83087 + 1.30330I
b = 0.831506 − 0.932621I
14.2431 + 0.5975I 11.31138 + 0.I
u = 0.538055 + 0.454086I
a = 2.48988 − 0.58955I
b = −0.551064 − 0.378134I
8.24921 − 1.23307I 10.61321 − 2.35997I
u = 0.538055 − 0.454086I
a = 2.48988 + 0.58955I
b = −0.551064 + 0.378134I
8.24921 + 1.23307I 10.61321 + 2.35997I
u = 0.347293 + 0.590519I
a = 0.19395 − 1.58608I
b = −0.859706 + 0.416440I
−1.72808 + 1.17860I −0.84051 − 1.97476I
u = 0.347293 − 0.590519I
a = 0.19395 + 1.58608I
b = −0.859706 − 0.416440I
−1.72808 − 1.17860I −0.84051 + 1.97476I
u = 1.043690 + 0.800779I
a = −0.495482 − 0.469030I
b = 0.892411 + 0.841271I
5.39547 − 3.53826I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.043690 − 0.800779I
a = −0.495482 + 0.469030I
b = 0.892411 − 0.841271I
5.39547 + 3.53826I 0
u = 0.087517 + 0.669029I
a = −0.577668 − 0.564351I
b = −1.323170 + 0.163226I
−2.60028 + 1.10335I 0.72944 − 5.14975I
u = 0.087517 − 0.669029I
a = −0.577668 + 0.564351I
b = −1.323170 − 0.163226I
−2.60028 − 1.10335I 0.72944 + 5.14975I
u = 0.807134 + 1.059990I
a = 0.655895 + 1.196400I
b = 0.925887 − 0.815812I
5.27984 + 2.65601I 0
u = 0.807134 − 1.059990I
a = 0.655895 − 1.196400I
b = 0.925887 + 0.815812I
5.27984 − 2.65601I 0
u = −0.118669 + 1.352730I
a = 0.692604 − 0.095605I
b = 0.673600 + 0.068678I
−3.90139 − 2.12133I 0
u = −0.118669 − 1.352730I
a = 0.692604 + 0.095605I
b = 0.673600 − 0.068678I
−3.90139 + 2.12133I 0
u = −1.061870 + 0.848054I
a = −0.596141 + 0.421898I
b = 1.020730 − 0.858173I
13.6503 + 6.0084I 0
u = −1.061870 − 0.848054I
a = −0.596141 − 0.421898I
b = 1.020730 + 0.858173I
13.6503 − 6.0084I 0
u = −0.453110 + 0.434147I
a = 1.81325 + 1.42213I
b = −0.588284 + 0.063513I
0.308257 + 0.438585I 7.79318 − 0.39703I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.453110 − 0.434147I
a = 1.81325 − 1.42213I
b = −0.588284 − 0.063513I
0.308257 − 0.438585I 7.79318 + 0.39703I
u = −0.851980 + 1.097830I
a = 0.462368 − 1.207420I
b = 1.062440 + 0.773956I
2.23629 − 6.46162I 0
u = −0.851980 − 1.097830I
a = 0.462368 + 1.207420I
b = 1.062440 − 0.773956I
2.23629 + 6.46162I 0
u = 0.890810 + 1.092080I
a = 0.358382 + 1.303470I
b = 1.16024 − 0.81354I
4.46235 + 10.57810I 0
u = 0.890810 − 1.092080I
a = 0.358382 − 1.303470I
b = 1.16024 + 0.81354I
4.46235 − 10.57810I 0
u = 0.369530 + 1.364470I
a = 0.633928 + 0.303473I
b = 0.736491 − 0.206857I
2.62006 + 5.04142I 0
u = 0.369530 − 1.364470I
a = 0.633928 − 0.303473I
b = 0.736491 + 0.206857I
2.62006 − 5.04142I 0
u = −0.91555 + 1.08098I
a = 0.29416 − 1.39483I
b = 1.22988 + 0.86093I
12.8768 − 13.1889I 0
u = −0.91555 − 1.08098I
a = 0.29416 + 1.39483I
b = 1.22988 − 0.86093I
12.8768 + 13.1889I 0
u = −0.500346
a = 6.58730
b = −1.11870
6.69758 29.8650
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.442850
a = 0.781638
b = 0.0358028
0.706374 14.0900
u = 0.326673
a = 11.6043
b = −1.02253
−0.833901 102.450
9
II.
I
u
2
= hb+1, u
5
−2u
4
+4u
3
−5u
2
+a+4u −3, u
6
−u
5
+3u
4
−2u
3
+2u
2
−u−1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
3
=
−u
5
+ 2u
4
− 4u
3
+ 5u
2
− 4u + 3
−1
a
6
=
1
−u
2
a
7
=
1
−u
2
a
2
=
−u
5
+ 2u
4
− 4u
3
+ 5u
2
− 4u + 2
−1
a
1
=
−1
0
a
11
=
u
u
a
4
=
−u
5
+ 2u
4
− 4u
3
+ 5u
2
− 4u + 3
−1
a
9
=
−u
u
3
+ u
a
12
=
u
4
+ u
2
− 1
−u
5
+ u
4
− 2u
3
+ u
2
− u − 1
a
8
=
−u
4
− u
2
+ 1
−u
4
− 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 7u
5
− 7u
4
+ 21u
3
− 17u
2
+ 20u − 12
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
6
u
6
c
4
(u + 1)
6
c
5
, c
10
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
7
, c
8
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
9
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
11
, c
12
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
6
y
6
c
5
, c
9
, c
10
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
c
7
, c
8
, c
11
c
12
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.873214
a = 1.31147
b = −1.00000
6.01515 5.96810
u = −0.138835 + 1.234450I
a = −0.631845 + 0.143944I
b = −1.00000
−4.60518 − 1.97241I −1.94905 + 2.83524I
u = −0.138835 − 1.234450I
a = −0.631845 − 0.143944I
b = −1.00000
−4.60518 + 1.97241I −1.94905 − 2.83524I
u = 0.408802 + 1.276380I
a = −0.453123 − 0.323434I
b = −1.00000
2.05064 + 4.59213I 3.43197 − 0.44648I
u = 0.408802 − 1.276380I
a = −0.453123 + 0.323434I
b = −1.00000
2.05064 − 4.59213I 3.43197 + 0.44648I
u = −0.413150
a = 5.85846
b = −1.00000
−0.906083 −24.9340
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
42
+ 13u
41
+ ··· + 804u + 1)
c
2
((u − 1)
6
)(u
42
− 7u
41
+ ··· − 32u + 1)
c
3
, c
6
u
6
(u
42
+ 5u
41
+ ··· − 256u + 64)
c
4
((u + 1)
6
)(u
42
− 7u
41
+ ··· − 32u + 1)
c
5
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
42
− 2u
41
+ ··· − u + 1)
c
7
, c
8
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
42
− 2u
41
+ ··· + u + 1)
c
9
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
42
− 2u
41
+ ··· − u + 1)
c
10
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
42
+ 14u
41
+ ··· + 5149u + 1583)
c
11
, c
12
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
42
− 2u
41
+ ··· + u + 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
42
+ 39y
41
+ ··· − 608644y + 1)
c
2
, c
4
((y −1)
6
)(y
42
− 13y
41
+ ··· − 804y + 1)
c
3
, c
6
y
6
(y
42
− 39y
41
+ ··· − 253952y + 4096)
c
5
, c
9
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
42
+ 10y
41
+ ··· − 7y + 1)
c
7
, c
8
, c
11
c
12
(y
6
− 7y
5
+ ··· − 5y + 1)(y
42
− 50y
41
+ ··· − 7y + 1)
c
10
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
42
− 26y
41
+ ··· − 65134235y + 2505889)
15
