12n
0653
(K12n
0653
)
A knot diagram
1
Linearized knot diagam
3 8 11 12 8 10 2 5 3 6 4 9
Solving Sequence
4,12 5,8
6 9 1 11 3 2 7 10
c
4
c
5
c
8
c
12
c
11
c
3
c
2
c
7
c
10
c
1
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h15u
19
+ 52u
18
+ ··· + b − 21, 21u
19
+ 75u
18
+ ··· + 2a − 23, u
20
+ 5u
19
+ ··· + u − 2i
I
u
2
= h2u
12
− 2u
11
− 11u
10
+ 8u
9
+ 23u
8
− 5u
7
− 22u
6
− 13u
5
+ 4u
4
+ 16u
3
+ 8u
2
+ b − u − 2,
2u
12
− 2u
11
− 12u
10
+ 9u
9
+ 28u
8
− 9u
7
− 31u
6
− 10u
5
+ 11u
4
+ 20u
3
+ 8u
2
+ a − 6u − 5,
u
13
− 2u
12
− 5u
11
+ 10u
10
+ 10u
9
− 16u
8
− 13u
7
+ 6u
6
+ 12u
5
+ 8u
4
− 4u
3
− 7u
2
− 2u + 1i
I
u
3
= hu
5
− u
4
− 2u
3
− au + u
2
+ b + u + 1, −u
5
a − 4u
5
+ 4u
3
a + u
4
+ 11u
3
+ a
2
− 3au + u
2
− 2a − 4u − 6,
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 45 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
=
h15u
19
+52u
18
+· · ·+b−21, 21u
19
+75u
18
+· · ·+2a−23, u
20
+5u
19
+· · ·+u−2i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
5
=
1
u
2
a
8
=
−10.5000u
19
− 37.5000u
18
+ ··· − 17.5000u + 11.5000
−15u
19
− 52u
18
+ ··· − 22u + 21
a
6
=
11
2
u
19
+
37
2
u
18
+ ··· +
11
2
u −
11
2
9u
19
+ 30u
18
+ ··· + 12u − 11
a
9
=
−
5
2
u
19
−
19
2
u
18
+ ··· −
7
2
u +
5
2
4u
19
+ 13u
18
+ ··· + 6u − 3
a
1
=
−
5
2
u
19
−
17
2
u
18
+ ··· −
7
2
u +
7
2
−6u
19
− 20u
18
+ ··· − 7u + 7
a
11
=
u
u
a
3
=
−u
2
+ 1
−u
2
a
2
=
5
2
u
19
+
19
2
u
18
+ ··· +
9
2
u −
7
2
3u
19
+ 12u
18
+ ··· + 7u − 5
a
7
=
−23u
19
− 80u
18
+ ··· − 34u + 27
−32u
19
− 110u
18
+ ··· − 45u + 42
a
10
=
−22.5000u
19
− 78.5000u
18
+ ··· − 33.5000u + 27.5000
−29u
19
− 100u
18
+ ··· − 41u + 39
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −15u
19
− 52u
18
+ 51u
17
+ 300u
16
− 63u
15
− 689u
14
+ 302u
13
+ 905u
12
− 1067u
11
−
777u
10
+ 1394u
9
+ 39u
8
− 722u
7
+ 587u
6
+ 181u
5
+ 7u
4
+ 174u
3
− 147u
2
− 9u + 12
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
+ 31u
19
+ ··· − 5u + 1
c
2
, c
7
, c
12
u
20
+ u
19
+ ··· − 3u − 1
c
3
, c
4
, c
11
u
20
+ 5u
19
+ ··· + u − 2
c
5
, c
8
u
20
− u
19
+ ··· − 7u + 1
c
6
, c
10
u
20
+ 14u
19
+ ··· + 608u + 64
c
9
u
20
− 26u
18
+ ··· − 494u − 599
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
− 103y
19
+ ··· − 159y + 1
c
2
, c
7
, c
12
y
20
− 31y
19
+ ··· + 5y + 1
c
3
, c
4
, c
11
y
20
− 21y
19
+ ··· − 61y + 4
c
5
, c
8
y
20
+ 29y
19
+ ··· − 55y + 1
c
6
, c
10
y
20
+ 6y
19
+ ··· − 33792y + 4096
c
9
y
20
− 52y
19
+ ··· + 2577254y + 358801
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.537648 + 0.858586I
a = 0.960669 − 0.049342I
b = −0.558866 − 0.798289I
−14.2098 + 2.1239I −11.72729 + 0.16245I
u = 0.537648 − 0.858586I
a = 0.960669 + 0.049342I
b = −0.558866 + 0.798289I
−14.2098 − 2.1239I −11.72729 − 0.16245I
u = 0.606622 + 0.819368I
a = −0.914544 − 0.419396I
b = 0.211143 + 1.003760I
−14.4361 − 7.6779I −11.74067 + 4.57873I
u = 0.606622 − 0.819368I
a = −0.914544 + 0.419396I
b = 0.211143 − 1.003760I
−14.4361 + 7.6779I −11.74067 − 4.57873I
u = −1.223290 + 0.229738I
a = −0.518734 − 0.448442I
b = −0.737585 − 0.429401I
−1.41396 + 1.68379I −7.75011 + 2.58763I
u = −1.223290 − 0.229738I
a = −0.518734 + 0.448442I
b = −0.737585 + 0.429401I
−1.41396 − 1.68379I −7.75011 − 2.58763I
u = −0.087774 + 0.628145I
a = 0.205193 + 0.654515I
b = 0.429141 − 0.071442I
2.02337 + 1.45900I −2.57228 − 5.20755I
u = −0.087774 − 0.628145I
a = 0.205193 − 0.654515I
b = 0.429141 + 0.071442I
2.02337 − 1.45900I −2.57228 + 5.20755I
u = 1.382330 + 0.213207I
a = −0.232569 + 0.103641I
b = 0.343585 − 0.093682I
−2.65671 − 4.43053I −10.17327 + 4.41788I
u = 1.382330 − 0.213207I
a = −0.232569 − 0.103641I
b = 0.343585 + 0.093682I
−2.65671 + 4.43053I −10.17327 − 4.41788I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.43646
a = 0.235964
b = −0.338953
−6.53090 −14.6160
u = −1.51134 + 0.02336I
a = −0.24536 − 2.15134I
b = −0.42108 − 3.24569I
−7.13768 + 2.65728I −12.42433 − 2.06390I
u = −1.51134 − 0.02336I
a = −0.24536 + 2.15134I
b = −0.42108 + 3.24569I
−7.13768 − 2.65728I −12.42433 + 2.06390I
u = 0.418033 + 0.095358I
a = −0.16881 + 1.84075I
b = 0.246100 − 0.753394I
−0.61706 − 2.23609I −3.44792 + 0.63650I
u = 0.418033 − 0.095358I
a = −0.16881 − 1.84075I
b = 0.246100 + 0.753394I
−0.61706 + 2.23609I −3.44792 − 0.63650I
u = −1.56523 + 0.31596I
a = 0.97491 − 1.39039I
b = 1.08666 − 2.48431I
18.4230 + 2.2284I −14.3978 − 0.8992I
u = −1.56523 − 0.31596I
a = 0.97491 + 1.39039I
b = 1.08666 + 2.48431I
18.4230 − 2.2284I −14.3978 + 0.8992I
u = −1.58137 + 0.27874I
a = −0.51960 + 1.97559I
b = −0.27100 + 3.26896I
17.8528 + 11.7492I −14.2038 − 4.7569I
u = −1.58137 − 0.27874I
a = −0.51960 − 1.97559I
b = −0.27100 − 3.26896I
17.8528 − 11.7492I −14.2038 + 4.7569I
u = −0.387724
a = −0.818265
b = −0.317261
−0.639359 −15.5090
6
II.
I
u
2
= h2u
12
−2u
11
+· · ·+b−2, 2u
12
−2u
11
+· · ·+a−5, u
13
−2u
12
+· · ·−2u+1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
5
=
1
u
2
a
8
=
−2u
12
+ 2u
11
+ ··· + 6u + 5
−2u
12
+ 2u
11
+ ··· + u + 2
a
6
=
3u
12
− 3u
11
+ ··· − 5u − 5
3u
12
− 2u
11
+ ··· + 17u
2
− 3
a
9
=
−2u
12
+ 3u
11
+ ··· + 9u + 5
−u
12
+ 2u
11
+ ··· + 3u + 1
a
1
=
u
12
− 3u
11
+ ··· − 17u − 7
−u
11
+ u
10
+ 5u
9
− 3u
8
− 10u
7
+ 9u
5
+ 7u
4
− 6u
2
− 4u − 1
a
11
=
u
u
a
3
=
−u
2
+ 1
−u
2
a
2
=
−u
12
− u
11
+ ··· − 15u − 4
−3u
12
+ u
11
+ ··· − 7u + 1
a
7
=
5u
12
− 6u
11
+ ··· − 10u − 9
5u
12
− 4u
11
+ ··· + u − 6
a
10
=
−3u
12
+ 3u
11
+ ··· + 5u + 5
−4u
12
+ 4u
11
+ ··· + u + 3
(ii) Obstruction class = 1
(iii) Cusp Shapes
= −4u
12
+ 9u
11
+ 17u
10
−41u
9
−27u
8
+ 56u
7
+ 32u
6
−11u
5
−30u
4
−29u
3
+ 8u
2
+ 12u −8
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
13
− 15u
12
+ ··· + 9u − 1
c
2
u
13
+ u
12
+ ··· + 3u − 1
c
3
, c
4
u
13
− 2u
12
+ ··· − 2u + 1
c
5
u
13
− u
12
+ ··· + u − 1
c
6
u
13
+ u
12
+ ··· + u + 1
c
7
, c
12
u
13
− u
12
+ ··· + 3u + 1
c
8
u
13
+ u
12
+ ··· + u + 1
c
9
u
13
− 6u
11
− 8u
10
+ 7u
8
+ 3u
7
+ 16u
6
− 2u
5
+ 14u
4
− 3u
3
+ 6u
2
+ 1
c
10
u
13
− u
12
+ ··· + u − 1
c
11
u
13
+ 2u
12
+ ··· − 2u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
13
− 31y
12
+ ··· − 15y −1
c
2
, c
7
, c
12
y
13
− 15y
12
+ ··· + 9y −1
c
3
, c
4
, c
11
y
13
− 14y
12
+ ··· + 18y −1
c
5
, c
8
y
13
+ 5y
12
+ ··· − 7y −1
c
6
, c
10
y
13
+ 7y
12
+ ··· − 5y −1
c
9
y
13
− 12y
12
+ ··· − 12y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.132140 + 0.168986I
a = 0.092976 + 0.325055I
b = −0.160192 − 0.352296I
−1.75574 + 2.59120I −12.09350 − 3.87877I
u = −1.132140 − 0.168986I
a = 0.092976 − 0.325055I
b = −0.160192 + 0.352296I
−1.75574 − 2.59120I −12.09350 + 3.87877I
u = −0.189605 + 0.771385I
a = 0.646314 + 0.286377I
b = −0.343451 + 0.444258I
0.707606 + 0.983665I −10.21762 − 1.58969I
u = −0.189605 − 0.771385I
a = 0.646314 − 0.286377I
b = −0.343451 − 0.444258I
0.707606 − 0.983665I −10.21762 + 1.58969I
u = 1.27456
a = −2.12141
b = −2.70385
−10.1403 −16.1260
u = −0.596279 + 0.393194I
a = −0.433017 + 0.867082I
b = −0.082733 − 0.687283I
−1.18520 + 2.76688I −11.52015 − 6.83060I
u = −0.596279 − 0.393194I
a = −0.433017 − 0.867082I
b = −0.082733 + 0.687283I
−1.18520 − 2.76688I −11.52015 + 6.83060I
u = 1.369650 + 0.339979I
a = 0.746659 + 0.696939I
b = 0.78571 + 1.20841I
−4.20868 − 5.03112I −13.5740 + 4.7480I
u = 1.369650 − 0.339979I
a = 0.746659 − 0.696939I
b = 0.78571 − 1.20841I
−4.20868 + 5.03112I −13.5740 − 4.7480I
u = −1.51524
a = −0.421908
b = 0.639294
−12.9834 −12.2240
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.53980 + 0.14093I
a = −0.10346 − 1.87763I
b = 0.10530 − 2.90577I
−8.24712 − 4.85635I −12.41808 + 4.09970I
u = 1.53980 − 0.14093I
a = −0.10346 + 1.87763I
b = 0.10530 + 2.90577I
−8.24712 + 4.85635I −12.41808 − 4.09970I
u = 0.257830
a = 5.64437
b = 1.45529
−6.71567 −5.00350
11
III. I
u
3
= hu
5
− u
4
− 2u
3
− au + u
2
+ b + u + 1, −u
5
a − 4u
5
+ · · · − 2a −
6, u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
5
=
1
u
2
a
8
=
a
−u
5
+ u
4
+ 2u
3
+ au − u
2
− u − 1
a
6
=
u
4
a + u
5
− u
3
a − u
4
− u
2
a − u
3
+ a − u
u
4
a − u
3
a − u
2
a + 2u
3
− 2u
2
+ a − 3u − 1
a
9
=
−u
5
+ u
4
− u
2
a + 2u
3
+ au − u
2
+ a − u − 1
−u
4
a − 2u
5
+ u
3
a + 2u
4
+ 3u
3
+ au − u
2
− 2u − 1
a
1
=
−3u
5
+ 2u
4
+ 8u
3
+ au − 2u
2
− a − 5u − 4
−2u
5
+ 2u
4
+ 4u
3
+ au − 2u
2
− 2u − 2
a
11
=
u
u
a
3
=
−u
2
+ 1
−u
2
a
2
=
−u
4
a − 3u
5
+ u
3
a + u
4
+ u
2
a + 7u
3
− a − 4u − 2
−u
4
a − 4u
5
+ u
3
a + 2u
4
+ 8u
3
+ au − u
2
− 3u − 3
a
7
=
−2u
4
a − 2u
5
+ 2u
3
a + 2u
4
+ 2u
2
a + 2u
3
− 2a + u
−2u
4
a + 2u
3
a + 2u
2
a − 4u
3
+ 4u
2
− 2a + 5u + 2
a
10
=
u
4
a + u
5
− u
3
a − u
4
− u
2
a − u
3
+ a
u
4
a − u
3
a − u
2
a + 2u
3
− 2u
2
+ a − 2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
3
− 8u − 18
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
+ 21u
11
+ ··· + 44976u + 9409
c
2
, c
7
, c
12
u
12
− u
11
+ ··· + 68u + 97
c
3
, c
4
, c
11
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
2
c
5
, c
8
u
12
+ 7u
11
+ ··· − 48u − 23
c
6
, c
10
(u − 1)
12
c
9
u
12
− u
11
+ ··· − 320u + 239
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
− 37y
11
+ ··· − 144089092y + 88529281
c
2
, c
7
, c
12
y
12
− 21y
11
+ ··· − 44976y + 9409
c
3
, c
4
, c
11
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
2
c
5
, c
8
y
12
+ 7y
11
+ ··· − 7180y + 529
c
6
, c
10
(y −1)
12
c
9
y
12
− 33y
11
+ ··· − 136816y + 57121
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.493180 + 0.575288I
a = 0.597504 − 0.159655I
b = −0.294522 + 1.202870I
−3.61949 + 1.97241I −12.57572 − 3.68478I
u = −0.493180 + 0.575288I
a = −1.45815 + 0.73808I
b = 0.202829 − 0.422475I
−3.61949 + 1.97241I −12.57572 − 3.68478I
u = −0.493180 − 0.575288I
a = 0.597504 + 0.159655I
b = −0.294522 − 1.202870I
−3.61949 − 1.97241I −12.57572 + 3.68478I
u = −0.493180 − 0.575288I
a = −1.45815 − 0.73808I
b = 0.202829 + 0.422475I
−3.61949 − 1.97241I −12.57572 + 3.68478I
u = 0.483672
a = −1.45315
b = −2.16590
−7.31859 −21.4170
u = 0.483672
a = 4.47804
b = 0.702848
−7.31859 −21.4170
u = 1.52087 + 0.16310I
a = −0.53855 − 1.90937I
b = −0.74685 − 3.39023I
−10.27530 − 4.59213I −16.5811 + 3.2048I
u = 1.52087 + 0.16310I
a = 0.72182 + 2.15174I
b = 0.50765 + 2.99173I
−10.27530 − 4.59213I −16.5811 + 3.2048I
u = 1.52087 − 0.16310I
a = −0.53855 + 1.90937I
b = −0.74685 + 3.39023I
−10.27530 + 4.59213I −16.5811 − 3.2048I
u = 1.52087 − 0.16310I
a = 0.72182 − 2.15174I
b = 0.50765 − 2.99173I
−10.27530 + 4.59213I −16.5811 − 3.2048I
15
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.53904
a = 1.22065
b = 3.24620
−14.2398 −20.2690
u = −1.53904
a = 2.10923
b = 1.87863
−14.2398 −20.2690
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
12
+ 21u
11
+ ··· + 44976u + 9409)(u
13
− 15u
12
+ ··· + 9u − 1)
· (u
20
+ 31u
19
+ ··· − 5u + 1)
c
2
(u
12
− u
11
+ ··· + 68u + 97)(u
13
+ u
12
+ ··· + 3u − 1)
· (u
20
+ u
19
+ ··· − 3u − 1)
c
3
, c
4
((u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
2
)(u
13
− 2u
12
+ ··· − 2u + 1)
· (u
20
+ 5u
19
+ ··· + u − 2)
c
5
(u
12
+ 7u
11
+ ··· − 48u − 23)(u
13
− u
12
+ ··· + u − 1)
· (u
20
− u
19
+ ··· − 7u + 1)
c
6
((u − 1)
12
)(u
13
+ u
12
+ ··· + u + 1)(u
20
+ 14u
19
+ ··· + 608u + 64)
c
7
, c
12
(u
12
− u
11
+ ··· + 68u + 97)(u
13
− u
12
+ ··· + 3u + 1)
· (u
20
+ u
19
+ ··· − 3u − 1)
c
8
(u
12
+ 7u
11
+ ··· − 48u − 23)(u
13
+ u
12
+ ··· + u + 1)
· (u
20
− u
19
+ ··· − 7u + 1)
c
9
(u
12
− u
11
+ ··· − 320u + 239)
· (u
13
− 6u
11
− 8u
10
+ 7u
8
+ 3u
7
+ 16u
6
− 2u
5
+ 14u
4
− 3u
3
+ 6u
2
+ 1)
· (u
20
− 26u
18
+ ··· − 494u − 599)
c
10
((u − 1)
12
)(u
13
− u
12
+ ··· + u − 1)(u
20
+ 14u
19
+ ··· + 608u + 64)
c
11
((u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
2
)(u
13
+ 2u
12
+ ··· − 2u − 1)
· (u
20
+ 5u
19
+ ··· + u − 2)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
12
− 37y
11
+ ··· − 144089092y + 88529281)
· (y
13
− 31y
12
+ ··· − 15y −1)(y
20
− 103y
19
+ ··· − 159y + 1)
c
2
, c
7
, c
12
(y
12
− 21y
11
+ ··· − 44976y + 9409)(y
13
− 15y
12
+ ··· + 9y −1)
· (y
20
− 31y
19
+ ··· + 5y + 1)
c
3
, c
4
, c
11
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
2
· (y
13
− 14y
12
+ ··· + 18y −1)(y
20
− 21y
19
+ ··· − 61y + 4)
c
5
, c
8
(y
12
+ 7y
11
+ ··· − 7180y + 529)(y
13
+ 5y
12
+ ··· − 7y −1)
· (y
20
+ 29y
19
+ ··· − 55y + 1)
c
6
, c
10
((y −1)
12
)(y
13
+ 7y
12
+ ··· − 5y −1)
· (y
20
+ 6y
19
+ ··· − 33792y + 4096)
c
9
(y
12
− 33y
11
+ ··· − 136816y + 57121)(y
13
− 12y
12
+ ··· − 12y −1)
· (y
20
− 52y
19
+ ··· + 2577254y + 358801)
18
