12n
0167
(K12n
0167
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 9 11 12 3 1 6 7 10
Solving Sequence
6,10
11 7 12 1
3,9
4 5 2 8
c
10
c
6
c
11
c
12
c
9
c
3
c
5
c
2
c
8
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
38
− 42u
36
+ ··· + b + 2, −u
38
− u
37
+ ··· + a − 2, u
39
+ 2u
38
+ ··· + 2u + 1i
I
u
2
= hu
4
− 2u
2
+ b, −u
5
+ u
4
+ 3u
3
− 2u
2
+ a − 2u − 1, u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
* 2 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
= h2u
38
−42u
36
+· · ·+b+2, −u
38
−u
37
+· · ·+a−2, u
39
+2u
38
+· · ·+2u+1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
12
=
−u
2
+ 1
u
4
− 2u
2
a
1
=
u
4
− 3u
2
+ 1
u
4
− 2u
2
a
3
=
u
38
+ u
37
+ ··· − 7u + 2
−2u
38
+ 42u
36
+ ··· − 5u − 2
a
9
=
u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1
u
8
− 4u
6
+ 4u
4
a
4
=
u
38
+ u
37
+ ··· − 8u + 1
−4u
38
+ 84u
36
+ ··· − 8u − 4
a
5
=
−u
17
+ 10u
15
− 39u
13
+ 74u
11
− 71u
9
+ 38u
7
− 18u
5
+ 4u
3
− u
−u
17
+ 9u
15
− 31u
13
+ 50u
11
− 37u
9
+ 12u
7
− 4u
5
+ u
a
2
=
u
38
+ u
37
+ ··· − 5u + 3
−u
38
+ 21u
36
+ ··· − 3u − 1
a
8
=
−u
3
+ 2u
u
5
− 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −8u
38
− 11u
37
+ ··· + 34u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
39
+ 11u
38
+ ··· − 5u + 1
c
2
, c
4
u
39
− 7u
38
+ ··· − 3u + 1
c
3
, c
8
u
39
+ u
38
+ ··· + 64u + 64
c
5
u
39
− 2u
38
+ ··· + 2u + 1
c
6
, c
7
, c
10
c
11
u
39
+ 2u
38
+ ··· + 2u + 1
c
9
, c
12
u
39
+ 8u
38
+ ··· + 70u − 7
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
39
+ 41y
38
+ ··· + 87y −1
c
2
, c
4
y
39
− 11y
38
+ ··· − 5y −1
c
3
, c
8
y
39
+ 39y
38
+ ··· − 24576y −4096
c
5
y
39
− 44y
38
+ ··· + 26y −1
c
6
, c
7
, c
10
c
11
y
39
− 44y
38
+ ··· + 26y −1
c
9
, c
12
y
39
+ 16y
38
+ ··· + 3318y −49
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.929287 + 0.081478I
a = −0.13426 + 1.82517I
b = −0.120150 + 0.122224I
7.67542 − 3.35786I 7.74116 + 3.09384I
u = −0.929287 − 0.081478I
a = −0.13426 − 1.82517I
b = −0.120150 − 0.122224I
7.67542 + 3.35786I 7.74116 − 3.09384I
u = 0.638465 + 0.572171I
a = 0.99563 + 1.99588I
b = 2.30175 + 0.94966I
3.77609 + 9.59779I 3.17812 − 7.93232I
u = 0.638465 − 0.572171I
a = 0.99563 − 1.99588I
b = 2.30175 − 0.94966I
3.77609 − 9.59779I 3.17812 + 7.93232I
u = 0.662133 + 0.524823I
a = −0.67104 − 1.69728I
b = −1.95714 − 0.96279I
4.90817 + 2.86742I 5.04494 − 3.58436I
u = 0.662133 − 0.524823I
a = −0.67104 + 1.69728I
b = −1.95714 + 0.96279I
4.90817 − 2.86742I 5.04494 + 3.58436I
u = −0.494019 + 0.600246I
a = 0.252653 + 0.393043I
b = 0.096843 + 0.148262I
−3.75312 − 2.04581I 4.95801 + 3.83439I
u = −0.494019 − 0.600246I
a = 0.252653 − 0.393043I
b = 0.096843 − 0.148262I
−3.75312 + 2.04581I 4.95801 − 3.83439I
u = −0.567091 + 0.480850I
a = −0.585871 + 0.875832I
b = −0.215752 + 0.266566I
−1.26346 − 3.63942I 1.82295 + 7.50238I
u = −0.567091 − 0.480850I
a = −0.585871 − 0.875832I
b = −0.215752 − 0.266566I
−1.26346 + 3.63942I 1.82295 − 7.50238I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.307625 + 0.630440I
a = −2.49493 − 0.52608I
b = −1.53447 + 0.98532I
2.80401 − 5.51652I 1.04879 + 2.33706I
u = 0.307625 − 0.630440I
a = −2.49493 + 0.52608I
b = −1.53447 − 0.98532I
2.80401 + 5.51652I 1.04879 − 2.33706I
u = 0.487298 + 0.473095I
a = −1.50995 + 1.72805I
b = 0.60093 + 2.21671I
−3.11331 + 1.66779I 1.80697 − 3.74196I
u = 0.487298 − 0.473095I
a = −1.50995 − 1.72805I
b = 0.60093 − 2.21671I
−3.11331 − 1.66779I 1.80697 + 3.74196I
u = 0.237194 + 0.597912I
a = 2.21111 + 0.47598I
b = 1.19865 − 0.81016I
3.66747 + 0.95204I 2.02608 − 2.37857I
u = 0.237194 − 0.597912I
a = 2.21111 − 0.47598I
b = 1.19865 + 0.81016I
3.66747 − 0.95204I 2.02608 + 2.37857I
u = −1.373730 + 0.070434I
a = 0.257891 − 1.073270I
b = 0.278657 + 1.104960I
7.90973 + 2.98789I 0
u = −1.373730 − 0.070434I
a = 0.257891 + 1.073270I
b = 0.278657 − 1.104960I
7.90973 − 2.98789I 0
u = −0.376548 + 0.452986I
a = 1.297760 − 0.214121I
b = 0.390433 − 0.135895I
−1.82437 + 0.31015I −1.43952 + 0.71227I
u = −0.376548 − 0.452986I
a = 1.297760 + 0.214121I
b = 0.390433 + 0.135895I
−1.82437 − 0.31015I −1.43952 − 0.71227I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.570048 + 0.105370I
a = 0.117467 − 0.126119I
b = −0.536626 − 0.310271I
0.990011 + 0.147928I 10.02163 − 0.80608I
u = 0.570048 − 0.105370I
a = 0.117467 + 0.126119I
b = −0.536626 + 0.310271I
0.990011 − 0.147928I 10.02163 + 0.80608I
u = 1.51500 + 0.17385I
a = −0.236905 + 0.112592I
b = −0.307436 + 0.170867I
2.85829 + 4.80629I 0
u = 1.51500 − 0.17385I
a = −0.236905 − 0.112592I
b = −0.307436 − 0.170867I
2.85829 − 4.80629I 0
u = 1.52238 + 0.09213I
a = −0.528996 − 0.528049I
b = −0.736761 − 0.687194I
4.57565 + 1.35883I 0
u = 1.52238 − 0.09213I
a = −0.528996 + 0.528049I
b = −0.736761 + 0.687194I
4.57565 − 1.35883I 0
u = −1.53584 + 0.12162I
a = 1.040540 + 0.300970I
b = −1.70605 + 2.33955I
3.67223 − 3.72431I 0
u = −1.53584 − 0.12162I
a = 1.040540 − 0.300970I
b = −1.70605 − 2.33955I
3.67223 + 3.72431I 0
u = −1.55793 + 0.04362I
a = −0.237922 − 0.022146I
b = 1.048690 − 0.750159I
8.23625 − 0.76574I 0
u = −1.55793 − 0.04362I
a = −0.237922 + 0.022146I
b = 1.048690 + 0.750159I
8.23625 + 0.76574I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.55813 + 0.13570I
a = 0.109432 + 0.550566I
b = 0.178775 + 0.723045I
5.88421 + 5.85782I 0
u = 1.55813 − 0.13570I
a = 0.109432 − 0.550566I
b = 0.178775 − 0.723045I
5.88421 − 5.85782I 0
u = −1.57931 + 0.17338I
a = 0.210755 + 1.250430I
b = −2.99968 + 0.82586I
11.2088 − 12.3506I 0
u = −1.57931 − 0.17338I
a = 0.210755 − 1.250430I
b = −2.99968 − 0.82586I
11.2088 + 12.3506I 0
u = −1.58740 + 0.15498I
a = −0.271413 − 1.006140I
b = 2.71217 − 1.00066I
12.48550 − 5.38089I 0
u = −1.58740 − 0.15498I
a = −0.271413 + 1.006140I
b = 2.71217 + 1.00066I
12.48550 + 5.38089I 0
u = 1.62512 + 0.01288I
a = 0.034213 + 1.205770I
b = 0.05087 + 1.57468I
16.3023 + 3.6362I 0
u = 1.62512 − 0.01288I
a = 0.034213 − 1.205770I
b = 0.05087 − 1.57468I
16.3023 − 3.6362I 0
u = −0.244498
a = 3.28769
b = 0.512609
−1.28163 −11.2860
8
II. I
u
2
=
hu
4
−2u
2
+b, −u
5
+u
4
+3u
3
−2u
2
+a−2u−1, u
6
−u
5
−3u
4
+2u
3
+2u
2
+u−1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
12
=
−u
2
+ 1
u
4
− 2u
2
a
1
=
u
4
− 3u
2
+ 1
u
4
− 2u
2
a
3
=
u
5
− u
4
− 3u
3
+ 2u
2
+ 2u + 1
−u
4
+ 2u
2
a
9
=
−u
3
+ 2u
u
5
− 3u
3
+ u
a
4
=
u
5
− u
4
− 3u
3
+ 2u
2
+ 2u + 1
−u
4
+ 2u
2
a
5
=
−u
4
+ 3u
2
− 1
−u
4
+ 2u
2
a
2
=
u
5
− 3u
3
− u
2
+ 2u + 2
0
a
8
=
−u
3
+ 2u
u
5
− 3u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
5
− u
4
− 14u
3
+ u
2
+ 14u + 6
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
8
u
6
c
4
(u + 1)
6
c
5
, c
9
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
6
, c
7
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
10
, c
11
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
12
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
8
y
6
c
5
, c
9
, c
12
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
c
6
, c
7
, c
10
c
11
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.493180 + 0.575288I
a = −0.858925 − 1.001920I
b = 0.138835 − 1.234450I
−4.60518 − 1.97241I −5.56070 + 3.48596I
u = −0.493180 − 0.575288I
a = −0.858925 + 1.001920I
b = 0.138835 + 1.234450I
−4.60518 + 1.97241I −5.56070 − 3.48596I
u = 0.483672
a = 2.06752
b = 0.413150
−0.906083 11.4460
u = 1.52087 + 0.16310I
a = 0.650045 − 0.069710I
b = −0.408802 − 1.276380I
2.05064 + 4.59213I −1.33400 − 2.48468I
u = 1.52087 − 0.16310I
a = 0.650045 + 0.069710I
b = −0.408802 + 1.276380I
2.05064 − 4.59213I −1.33400 + 2.48468I
u = −1.53904
a = −0.649754
b = −0.873214
6.01515 6.34350
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
39
+ 11u
38
+ ··· − 5u + 1)
c
2
((u − 1)
6
)(u
39
− 7u
38
+ ··· − 3u + 1)
c
3
, c
8
u
6
(u
39
+ u
38
+ ··· + 64u + 64)
c
4
((u + 1)
6
)(u
39
− 7u
38
+ ··· − 3u + 1)
c
5
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
39
− 2u
38
+ ··· + 2u + 1)
c
6
, c
7
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
39
+ 2u
38
+ ··· + 2u + 1)
c
9
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
39
+ 8u
38
+ ··· + 70u − 7)
c
10
, c
11
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
39
+ 2u
38
+ ··· + 2u + 1)
c
12
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
39
+ 8u
38
+ ··· + 70u − 7)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
39
+ 41y
38
+ ··· + 87y −1)
c
2
, c
4
((y −1)
6
)(y
39
− 11y
38
+ ··· − 5y −1)
c
3
, c
8
y
6
(y
39
+ 39y
38
+ ··· − 24576y −4096)
c
5
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
39
− 44y
38
+ ··· + 26y −1)
c
6
, c
7
, c
10
c
11
(y
6
− 7y
5
+ ··· − 5y + 1)(y
39
− 44y
38
+ ··· + 26y −1)
c
9
, c
12
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
39
+ 16y
38
+ ··· + 3318y −49)
14
