12a
1164
(K12a
1164
)
A knot diagram
1
Linearized knot diagam
4 9 8 12 1 10 11 3 2 7 6 5
Solving Sequence
4,12
5 1 2
6,8
3 9 11 7 10
c
4
c
12
c
1
c
5
c
3
c
8
c
11
c
7
c
10
c
2
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
19
+ u
18
+ ··· + 2b − 1, u
6
− 3u
4
+ 2u
2
+ a + 1, u
21
+ u
20
+ ··· + u + 1i
I
u
2
= h−1056266u
33
− 534744u
32
+ ··· + 2309809b − 234638,
− 619436u
33
− 841682u
32
+ ··· + 6929427a − 2733846, u
34
+ u
33
+ ··· + 6u − 3i
I
u
3
= hb
2
+ 2, a + 1, u − 1i
I
u
4
= hb, a + 1, u + 1i
* 4 irreducible components of dim
C
= 0, with total 58 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
= hu
19
+ u
18
+ · · · + 2b − 1, u
6
− 3u
4
+ 2u
2
+ a + 1, u
21
+ u
20
+ · · · + u + 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
5
=
1
u
2
a
1
=
−u
−u
3
+ u
a
2
=
u
3
− 2u
−u
3
+ u
a
6
=
−u
2
+ 1
−u
4
+ 2u
2
a
8
=
−u
6
+ 3u
4
− 2u
2
− 1
−
1
2
u
19
−
1
2
u
18
+ ··· +
1
2
u +
1
2
a
3
=
1
2
u
19
+
1
2
u
18
+ ··· −
1
2
u +
1
2
1
2
u
20
− 5u
18
+ ··· − u −
1
2
a
9
=
−
1
2
u
20
−
1
2
u
19
+ ··· +
1
2
u
2
+
5
2
u
1
2
u
20
+
1
2
u
19
+ ··· −
1
2
u
2
+
1
2
u
a
11
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
7
=
u
4
− u
2
− 1
−
1
2
u
19
−
1
2
u
18
+ ··· +
1
2
u +
1
2
a
10
=
−u
3
+ 2u
1
2
u
20
+
1
2
u
19
+ ··· −
1
2
u
2
+
1
2
u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −u
20
+ u
19
+ 13u
18
− 10u
17
− 67u
16
+ 44u
15
+ 175u
14
− 107u
13
− 225u
12
+ 145u
11
+
63u
10
− 83u
9
+ 168u
8
− 36u
7
− 146u
6
+ 70u
5
− 25u
4
− 11u
3
+ 41u
2
− 17u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
21
− 3u
20
+ ··· + 16u
2
− 16
c
2
, c
3
, c
8
c
9
u
21
+ 3u
20
+ ··· − 4u − 2
c
4
, c
5
, c
6
c
7
, c
10
, c
12
u
21
+ u
20
+ ··· + u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
21
+ 11y
20
+ ··· + 512y −256
c
2
, c
3
, c
8
c
9
y
21
+ 23y
20
+ ··· − 32y −4
c
4
, c
5
, c
6
c
7
, c
10
, c
12
y
21
− 21y
20
+ ··· − 3y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.121015 + 0.802604I
a = 1.51299 − 1.34586I
b = −0.16702 + 1.51542I
−2.09551 − 4.69375I −3.63503 + 3.84735I
u = 0.121015 − 0.802604I
a = 1.51299 + 1.34586I
b = −0.16702 − 1.51542I
−2.09551 + 4.69375I −3.63503 − 3.84735I
u = −0.040007 + 0.789380I
a = 1.62258 + 0.43481I
b = −0.588267 − 0.491565I
4.51526 + 2.01021I 0.41448 − 3.78371I
u = −0.040007 − 0.789380I
a = 1.62258 − 0.43481I
b = −0.588267 + 0.491565I
4.51526 − 2.01021I 0.41448 + 3.78371I
u = 1.283400 + 0.250591I
a = 0.110044 + 0.250523I
b = −0.205916 − 1.377950I
−9.03034 − 2.72457I −12.33705 + 3.20097I
u = 1.283400 − 0.250591I
a = 0.110044 − 0.250523I
b = −0.205916 + 1.377950I
−9.03034 + 2.72457I −12.33705 − 3.20097I
u = −1.35078
a = −0.736116
b = 0.673734
−7.60400 −10.4800
u = −1.319980 + 0.321329I
a = 0.757736 − 0.419310I
b = −0.688597 + 0.357279I
−3.53789 + 5.93688I −7.80246 − 3.12775I
u = −1.319980 − 0.321329I
a = 0.757736 + 0.419310I
b = −0.688597 − 0.357279I
−3.53789 − 5.93688I −7.80246 + 3.12775I
u = 1.388650 + 0.093259I
a = −0.673057 − 0.380782I
b = 0.353596 − 0.874613I
−10.38940 − 3.55849I −14.4570 + 4.3859I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.388650 − 0.093259I
a = −0.673057 + 0.380782I
b = 0.353596 + 0.874613I
−10.38940 + 3.55849I −14.4570 − 4.3859I
u = 1.353780 + 0.356484I
a = 1.326430 + 0.422608I
b = −0.624029 + 0.625071I
−4.33607 − 10.34760I −9.08482 + 8.30410I
u = 1.353780 − 0.356484I
a = 1.326430 − 0.422608I
b = −0.624029 − 0.625071I
−4.33607 + 10.34760I −9.08482 − 8.30410I
u = −1.38701 + 0.37866I
a = 1.88840 − 0.29011I
b = −0.19424 − 1.57233I
−11.6562 + 13.3679I −11.96534 − 7.27527I
u = −1.38701 − 0.37866I
a = 1.88840 + 0.29011I
b = −0.19424 + 1.57233I
−11.6562 − 13.3679I −11.96534 + 7.27527I
u = 0.445437 + 0.325177I
a = −1.38881 − 0.40146I
b = 0.02269 − 1.49213I
−6.40486 − 1.31691I −7.01863 + 5.01587I
u = 0.445437 − 0.325177I
a = −1.38881 + 0.40146I
b = 0.02269 + 1.49213I
−6.40486 + 1.31691I −7.01863 − 5.01587I
u = −1.47176 + 0.13523I
a = −0.818744 + 1.112050I
b = 0.06951 + 1.62601I
−18.9337 + 4.9859I −15.7556 − 3.2060I
u = −1.47176 − 0.13523I
a = −0.818744 − 1.112050I
b = 0.06951 − 1.62601I
−18.9337 − 4.9859I −15.7556 + 3.2060I
u = −0.198124 + 0.264706I
a = −0.969512 + 0.228317I
b = 0.185405 + 0.382871I
−0.126670 + 0.730510I −4.11837 − 9.53132I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.198124 − 0.264706I
a = −0.969512 − 0.228317I
b = 0.185405 − 0.382871I
−0.126670 − 0.730510I −4.11837 + 9.53132I
7
II.
I
u
2
= h−1.06 × 10
6
u
33
− 5.35 × 10
5
u
32
+ · · · + 2.31 × 10
6
b − 2.35 × 10
5
, −6.19 ×
10
5
u
33
− 8.42× 10
5
u
32
+ · · · + 6.93 × 10
6
a −2.73 × 10
6
, u
34
+ u
33
+ · · · + 6u − 3i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
5
=
1
u
2
a
1
=
−u
−u
3
+ u
a
2
=
u
3
− 2u
−u
3
+ u
a
6
=
−u
2
+ 1
−u
4
+ 2u
2
a
8
=
0.0893921u
33
+ 0.121465u
32
+ ··· + 3.10263u + 0.394527
0.457296u
33
+ 0.231510u
32
+ ··· − 1.08414u + 0.101583
a
3
=
−0.408040u
33
− 0.237636u
32
+ ··· + 4.75198u − 0.453759
−0.180498u
33
+ 0.325687u
32
+ ··· − 2.48353u + 0.264153
a
9
=
−0.0764438u
33
− 0.0671149u
32
+ ··· − 1.96468u + 3.75923
−0.151130u
33
− 0.0265009u
32
+ ··· + 1.29755u − 1.23819
a
11
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
7
=
−0.328926u
33
− 0.0100975u
32
+ ··· + 5.68192u − 0.510619
0.418318u
33
+ 0.131562u
32
+ ··· − 2.57929u − 0.0948537
a
10
=
0.350446u
33
+ 0.00787828u
32
+ ··· + 1.10883u + 1.78222
−0.352234u
33
− 0.131296u
32
+ ··· + 0.866235u − 0.633064
(ii) Obstruction class = −1
(iii) Cusp Shapes =
113576
2309809
u
33
+
5887808
2309809
u
32
+ ··· −
11535388
2309809
u −
230526
2309809
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
(u
17
− 3u
16
+ ··· + 9u − 3)
2
c
2
, c
3
, c
8
c
9
(u
17
− u
16
+ ··· + u + 1)
2
c
4
, c
5
, c
6
c
7
, c
10
, c
12
u
34
+ u
33
+ ··· + 6u − 3
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
(y
17
+ 11y
16
+ ··· + 57y −9)
2
c
2
, c
3
, c
8
c
9
(y
17
+ 19y
16
+ ··· + y −1)
2
c
4
, c
5
, c
6
c
7
, c
10
, c
12
y
34
− 25y
33
+ ··· − 48y + 9
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.180995 + 0.883653I
a = −1.35621 + 1.29271I
b = 0.17426 − 1.55100I
−6.70220 − 8.83664I −8.37368 + 5.87120I
u = 0.180995 − 0.883653I
a = −1.35621 − 1.29271I
b = 0.17426 + 1.55100I
−6.70220 + 8.83664I −8.37368 − 5.87120I
u = 0.595797 + 0.672047I
a = 0.993055 − 0.599481I
b = −0.02780 + 1.57600I
−12.06090 − 2.39923I −12.86600 + 3.27109I
u = 0.595797 − 0.672047I
a = 0.993055 + 0.599481I
b = −0.02780 − 1.57600I
−12.06090 + 2.39923I −12.86600 − 3.27109I
u = −1.13369
a = 0.407824
b = −0.387802
−2.28510 −1.13090
u = −0.136716 + 0.824881I
a = −1.58020 − 0.51459I
b = 0.580614 + 0.569922I
0.35577 + 6.09306I −4.70703 − 6.87425I
u = −0.136716 − 0.824881I
a = −1.58020 + 0.51459I
b = 0.580614 − 0.569922I
0.35577 − 6.09306I −4.70703 + 6.87425I
u = −1.101130 + 0.389395I
a = 0.509094 − 0.606456I
b = −0.488571 + 0.501958I
−2.59185 − 1.70542I −7.89077 + 4.02096I
u = −1.101130 − 0.389395I
a = 0.509094 + 0.606456I
b = −0.488571 − 0.501958I
−2.59185 + 1.70542I −7.89077 − 4.02096I
u = 1.128290 + 0.347386I
a = 0.098642 − 0.298753I
b = 0.14171 + 1.46572I
−5.15765 + 0.50801I −6.42549 + 0.23246I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.128290 − 0.347386I
a = 0.098642 + 0.298753I
b = 0.14171 − 1.46572I
−5.15765 − 0.50801I −6.42549 − 0.23246I
u = 1.080650 + 0.504832I
a = −0.224516 + 0.457220I
b = −0.13662 − 1.53895I
−9.44087 + 3.91820I −11.59784 − 2.39256I
u = 1.080650 − 0.504832I
a = −0.224516 − 0.457220I
b = −0.13662 + 1.53895I
−9.44087 − 3.91820I −11.59784 + 2.39256I
u = 0.078456 + 0.750182I
a = −1.69151 − 0.36066I
b = 0.601563 + 0.400803I
0.85249 − 2.05778I −2.98070 + 0.37816I
u = 0.078456 − 0.750182I
a = −1.69151 + 0.36066I
b = 0.601563 − 0.400803I
0.85249 + 2.05778I −2.98070 − 0.37816I
u = 1.213590 + 0.290321I
a = 1.38485 + 0.75199I
b = −0.488571 + 0.501958I
−2.59185 − 1.70542I −7.89077 + 4.02096I
u = 1.213590 − 0.290321I
a = 1.38485 − 0.75199I
b = −0.488571 − 0.501958I
−2.59185 + 1.70542I −7.89077 − 4.02096I
u = 1.260460 + 0.061296I
a = 0.428262 + 0.963327I
b = −0.151255 + 0.679822I
−4.41315 − 1.83062I −11.59303 + 5.22267I
u = 1.260460 − 0.061296I
a = 0.428262 − 0.963327I
b = −0.151255 − 0.679822I
−4.41315 + 1.83062I −11.59303 − 5.22267I
u = −1.230380 + 0.338033I
a = −0.652848 + 0.475134I
b = 0.601563 − 0.400803I
0.85249 + 2.05778I −2.98070 − 0.37816I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.230380 − 0.338033I
a = −0.652848 − 0.475134I
b = 0.601563 + 0.400803I
0.85249 − 2.05778I −2.98070 + 0.37816I
u = −0.502282 + 0.483544I
a = 0.782134 + 0.681918I
b = −0.151255 − 0.679822I
−4.41315 + 1.83062I −11.59303 − 5.22267I
u = −0.502282 − 0.483544I
a = 0.782134 − 0.681918I
b = −0.151255 + 0.679822I
−4.41315 − 1.83062I −11.59303 + 5.22267I
u = 1.293540 + 0.343405I
a = −1.35895 − 0.53387I
b = 0.580614 − 0.569922I
0.35577 − 6.09306I −4.70703 + 6.87425I
u = 1.293540 − 0.343405I
a = −1.35895 + 0.53387I
b = 0.580614 + 0.569922I
0.35577 + 6.09306I −4.70703 − 6.87425I
u = 0.043766 + 0.657258I
a = −1.87701 + 1.49405I
b = 0.14171 − 1.46572I
−5.15765 − 0.50801I −6.42549 − 0.23246I
u = 0.043766 − 0.657258I
a = −1.87701 − 1.49405I
b = 0.14171 + 1.46572I
−5.15765 + 0.50801I −6.42549 + 0.23246I
u = −1.312420 + 0.277525I
a = 2.25755 − 0.93271I
b = −0.13662 − 1.53895I
−9.44087 + 3.91820I −11.59784 − 2.39256I
u = −1.312420 − 0.277525I
a = 2.25755 + 0.93271I
b = −0.13662 + 1.53895I
−9.44087 − 3.91820I −11.59784 + 2.39256I
u = −1.370680 + 0.056095I
a = 0.56793 − 2.07715I
b = −0.02780 − 1.57600I
−12.06090 + 2.39923I −12.86600 − 3.27109I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.370680 − 0.056095I
a = 0.56793 + 2.07715I
b = −0.02780 + 1.57600I
−12.06090 − 2.39923I −12.86600 + 3.27109I
u = −1.343580 + 0.346101I
a = −2.09386 + 0.46604I
b = 0.17426 + 1.55100I
−6.70220 + 8.83664I −8.37368 − 5.87120I
u = −1.343580 − 0.346101I
a = −2.09386 − 0.46604I
b = 0.17426 − 1.55100I
−6.70220 − 8.83664I −8.37368 + 5.87120I
u = 0.376966
a = 2.21933
b = −0.387802
−2.28510 −1.13090
14
III. I
u
3
= hb
2
+ 2, a + 1, u − 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
1
a
5
=
1
1
a
1
=
−1
0
a
2
=
−1
0
a
6
=
0
1
a
8
=
−1
b
a
3
=
−b + 1
−2
a
9
=
b + 1
−b
a
11
=
0
1
a
7
=
−1
b + 1
a
10
=
1
−b
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
2
c
2
, c
3
, c
8
c
9
u
2
+ 2
c
4
, c
5
, c
10
(u − 1)
2
c
6
, c
7
, c
12
(u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
2
c
2
, c
3
, c
8
c
9
(y + 2)
2
c
4
, c
5
, c
6
c
7
, c
10
, c
12
(y −1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = 1.414210I
−8.22467 −12.0000
u = 1.00000
a = −1.00000
b = − 1.414210I
−8.22467 −12.0000
18
IV. I
u
4
= hb, a + 1, u + 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
−1
a
5
=
1
1
a
1
=
1
0
a
2
=
1
0
a
6
=
0
1
a
8
=
−1
0
a
3
=
1
0
a
9
=
−1
0
a
11
=
0
−1
a
7
=
−1
1
a
10
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
8
, c
9
, c
11
u
c
4
, c
5
, c
10
u + 1
c
6
, c
7
, c
12
u − 1
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
8
, c
9
, c
11
y
c
4
, c
5
, c
6
c
7
, c
10
, c
12
y −1
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
−3.28987 −12.0000
22
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
11
u
3
(u
17
− 3u
16
+ ··· + 9u − 3)
2
(u
21
− 3u
20
+ ··· + 16u
2
− 16)
c
2
, c
3
, c
8
c
9
u(u
2
+ 2)(u
17
− u
16
+ ··· + u + 1)
2
(u
21
+ 3u
20
+ ··· − 4u − 2)
c
4
, c
5
, c
10
((u − 1)
2
)(u + 1)(u
21
+ u
20
+ ··· + u + 1)(u
34
+ u
33
+ ··· + 6u − 3)
c
6
, c
7
, c
12
(u − 1)(u + 1)
2
(u
21
+ u
20
+ ··· + u + 1)(u
34
+ u
33
+ ··· + 6u − 3)
23
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
3
(y
17
+ 11y
16
+ ··· + 57y −9)
2
(y
21
+ 11y
20
+ ··· + 512y −256)
c
2
, c
3
, c
8
c
9
y(y + 2)
2
(y
17
+ 19y
16
+ ··· + y −1)
2
(y
21
+ 23y
20
+ ··· − 32y −4)
c
4
, c
5
, c
6
c
7
, c
10
, c
12
((y −1)
3
)(y
21
− 21y
20
+ ··· − 3y −1)(y
34
− 25y
33
+ ··· − 48y + 9)
24
