12n
0374
(K12n
0374
)
A knot diagram
1
Linearized knot diagam
3 6 8 10 2 11 3 12 4 6 9 7
Solving Sequence
6,10
11
3,7
8 2 1 5 4 9 12
c
10
c
6
c
7
c
2
c
1
c
5
c
4
c
9
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−301149306547334u
15
− 405785743495567u
14
+ ··· + 5460935417391053b − 2324129620964575,
2521976827730410u
15
+ 4134029936992622u
14
+ ··· + 5460935417391053a − 8351550389039156,
u
16
+ 2u
15
+ ··· + u − 1i
I
u
2
= hu
9
+ u
8
− 2u
7
+ 2u
6
+ 3u
5
− 6u
4
+ 2u
2
+ b − 2u − 1,
u
10
− 4u
8
+ 2u
7
+ 4u
6
− 6u
5
+ u
4
+ 4u
3
− u
2
+ a − u + 1,
u
11
+ u
10
− 3u
9
+ 5u
7
− 4u
6
− 4u
5
+ 4u
4
+ u
3
− 3u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 27 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−3.01×10
14
u
15
−4.06×10
14
u
14
+· · ·+5.46×10
15
b−2.32×10
15
, 2.52×
10
15
u
15
+4.13×10
15
u
14
+· · ·+5.46×10
15
a−8.35×10
15
, u
16
+2u
15
+· · ·+u−1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
3
=
−0.461821u
15
− 0.757019u
14
+ ··· + 1.13267u + 1.52933
0.0551461u
15
+ 0.0743070u
14
+ ··· − 1.30799u + 0.425592
a
7
=
−u
−u
3
+ u
a
8
=
−1.16646u
15
− 2.14400u
14
+ ··· − 0.648553u + 1.23228
0.142447u
15
+ 0.241985u
14
+ ··· − 2.03492u + 0.447884
a
2
=
−0.461821u
15
− 0.757019u
14
+ ··· + 1.13267u + 1.52933
0.0215283u
15
+ 0.0644709u
14
+ ··· − 0.679549u + 0.258968
a
1
=
1.49486u
15
+ 2.51422u
14
+ ··· − 4.17011u − 0.263898
−0.226698u
15
− 0.0915071u
14
+ ··· + 4.12577u − 0.807954
a
5
=
1.28737u
15
+ 2.32151u
14
+ ··· − 1.70682u − 1.04336
−0.120918u
15
− 0.177515u
14
+ ··· + 2.35537u − 0.188916
a
4
=
1.16646u
15
+ 2.14400u
14
+ ··· + 0.648553u − 1.23228
−0.120918u
15
− 0.177515u
14
+ ··· + 2.35537u − 0.188916
a
9
=
0.802633u
15
+ 1.98845u
14
+ ··· + 8.96383u − 3.11463
−0.352889u
15
− 0.635726u
14
+ ··· + 0.487447u + 0.504102
a
12
=
1.38603u
15
+ 2.45156u
14
+ ··· − 2.59193u − 0.500660
−0.149090u
15
− 0.0711014u
14
+ ··· + 2.28378u − 0.416208
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
9739656845787569
5460935417391053
u
15
−
17595322506035493
5460935417391053
u
14
+ ···+
16486080638841760
5460935417391053
u −
93508315704564998
5460935417391053
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
− 22u
15
+ ··· + 28u + 1
c
2
, c
5
u
16
+ 4u
15
+ ··· − 2u − 1
c
3
, c
7
u
16
− 14u
15
+ ··· + 152u
2
− 32
c
4
, c
9
u
16
+ u
15
+ ··· − 184u − 85
c
6
, c
10
u
16
− 2u
15
+ ··· − u − 1
c
8
, c
11
u
16
− 2u
15
+ ··· + 2u + 1
c
12
u
16
+ 8u
15
+ ··· − 17328u − 2417
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
+ 14y
15
+ ··· − 540y + 1
c
2
, c
5
y
16
+ 22y
15
+ ··· − 28y + 1
c
3
, c
7
y
16
− 14y
15
+ ··· − 9728y + 1024
c
4
, c
9
y
16
+ 15y
15
+ ··· − 36576y + 7225
c
6
, c
10
y
16
+ 20y
15
+ ··· − 9y + 1
c
8
, c
11
y
16
+ 10y
15
+ ··· − 14y + 1
c
12
y
16
+ 38y
15
+ ··· − 113705856y + 5841889
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.709961 + 0.302018I
a = 0.067289 − 0.371265I
b = −1.56748 + 0.56103I
−2.10600 − 5.42693I −16.6708 + 2.3025I
u = 0.709961 − 0.302018I
a = 0.067289 + 0.371265I
b = −1.56748 − 0.56103I
−2.10600 + 5.42693I −16.6708 − 2.3025I
u = −0.624053 + 0.414470I
a = 0.036931 + 0.619361I
b = −0.268149 + 0.147042I
2.12989 + 2.00985I −8.08247 − 3.69887I
u = −0.624053 − 0.414470I
a = 0.036931 − 0.619361I
b = −0.268149 − 0.147042I
2.12989 − 2.00985I −8.08247 + 3.69887I
u = −0.603403
a = 0.346440
b = 1.97720
−5.59033 −13.4830
u = 0.365789 + 0.462196I
a = 0.36041 − 1.55398I
b = 0.667180 + 0.526239I
−0.228166 + 0.909617I −12.89785 − 1.13498I
u = 0.365789 − 0.462196I
a = 0.36041 + 1.55398I
b = 0.667180 − 0.526239I
−0.228166 − 0.909617I −12.89785 + 1.13498I
u = −0.222804 + 0.369878I
a = 1.57769 + 3.86659I
b = 0.82840 − 1.19204I
−8.61028 − 1.60803I −19.3772 + 7.1837I
u = −0.222804 − 0.369878I
a = 1.57769 − 3.86659I
b = 0.82840 + 1.19204I
−8.61028 + 1.60803I −19.3772 − 7.1837I
u = 0.325252
a = 0.778005
b = 0.326406
−0.534698 −18.5610
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.14699 + 2.11502I
a = 1.050490 + 0.062469I
b = −5.41647 + 5.13168I
5.95217 − 5.11511I −14.9917 + 2.0744I
u = 1.14699 − 2.11502I
a = 1.050490 − 0.062469I
b = −5.41647 − 5.13168I
5.95217 + 5.11511I −14.9917 − 2.0744I
u = −0.86624 + 2.26332I
a = −1.018240 + 0.164220I
b = 6.83232 + 3.44115I
10.23600 + 0.21137I −11.36887 + 0.55661I
u = −0.86624 − 2.26332I
a = −1.018240 − 0.164220I
b = 6.83232 − 3.44115I
10.23600 − 0.21137I −11.36887 − 0.55661I
u = −1.37057 + 2.24654I
a = −1.136800 + 0.049742I
b = 5.77240 + 7.05674I
9.6708 + 10.3555I −12.00000 − 4.83541I
u = −1.37057 − 2.24654I
a = −1.136800 − 0.049742I
b = 5.77240 − 7.05674I
9.6708 − 10.3555I −12.00000 + 4.83541I
6
II.
I
u
2
= hu
9
+ u
8
+ · · · + b − 1, u
10
− 4u
8
+ · · · + a + 1, u
11
+ u
10
+ · · · − 3u
2
+ 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
3
=
−u
10
+ 4u
8
− 2u
7
− 4u
6
+ 6u
5
− u
4
− 4u
3
+ u
2
+ u − 1
−u
9
− u
8
+ 2u
7
− 2u
6
− 3u
5
+ 6u
4
− 2u
2
+ 2u + 1
a
7
=
−u
−u
3
+ u
a
8
=
−u
9
− u
8
+ 3u
7
− 5u
5
+ 3u
4
+ 3u
3
− 2u
2
− u + 1
2u
9
+ 2u
8
− 5u
7
+ 6u
5
− 6u
4
− 3u
3
+ 2u
2
− u − 1
a
2
=
−u
10
+ 4u
8
− 2u
7
− 4u
6
+ 6u
5
− u
4
− 4u
3
+ u
2
+ u − 1
−2u
9
− 2u
8
+ 5u
7
− u
6
− 7u
5
+ 8u
4
+ 3u
3
− 4u
2
+ u + 2
a
1
=
u
9
+ u
8
− 3u
7
− u
6
+ 4u
5
− u
4
− 2u
3
−2u
9
− 3u
8
+ 4u
7
+ 3u
6
− 4u
5
+ 2u
4
+ u
3
+ u
2
+ u
a
5
=
u
9
+ u
8
− 3u
7
− u
6
+ 4u
5
− u
4
− 3u
3
+ 2u
u
6
+ u
5
− 2u
4
+ 2u
2
− u − 1
a
4
=
u
9
+ u
8
− 3u
7
+ 5u
5
− 3u
4
− 3u
3
+ 2u
2
+ u − 1
u
6
+ u
5
− 2u
4
+ 2u
2
− u − 1
a
9
=
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
− 3u
2
+ 2
u
10
+ u
9
− 3u
8
− u
7
+ 4u
6
− 2u
5
− 3u
4
+ 2u
3
− u
a
12
=
u
4
− 2u
2
+ 1
−u
9
− u
8
+ 2u
7
− 2u
5
+ 2u
4
+ u
2
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 4u
10
+ 10u
9
− 2u
8
− 14u
7
+ 9u
6
+ 15u
5
− 26u
4
− 23u
3
+ 22u
2
− 2u − 29
7
(iv) u-Polynomials at the component
8
Crossings u-Polynomials at each crossing
c
1
u
11
− 9u
10
+ ··· + 5u − 1
c
2
u
11
+ 3u
10
− 6u
8
− 2u
7
+ 2u
6
+ 3u
5
+ 4u
4
+ 2u
3
− 2u
2
− 3u − 1
c
3
u
11
+ 2u
10
+ ··· − 5u + 1
c
4
u
11
− 4u
9
− 6u
8
+ 10u
7
+ 23u
6
− 10u
5
− 23u
4
− 2u
3
+ 16u
2
− 3u − 1
c
5
u
11
− 3u
10
+ 6u
8
− 2u
7
− 2u
6
+ 3u
5
− 4u
4
+ 2u
3
+ 2u
2
− 3u + 1
c
6
u
11
− u
10
− 3u
9
+ 5u
7
+ 4u
6
− 4u
5
− 4u
4
+ u
3
+ 3u
2
− 1
c
7
u
11
− 2u
10
+ ··· − 5u − 1
c
8
u
11
− 3u
10
+ ··· − u + 1
c
9
u
11
− 4u
9
+ 6u
8
+ 10u
7
− 23u
6
− 10u
5
+ 23u
4
− 2u
3
− 16u
2
− 3u + 1
c
10
u
11
+ u
10
− 3u
9
+ 5u
7
− 4u
6
− 4u
5
+ 4u
4
+ u
3
− 3u
2
+ 1
c
11
u
11
+ 3u
10
+ ··· − u − 1
c
12
u
11
− 7u
10
+ ··· − 85u + 29
9
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
11
− 17y
10
+ ··· + 9y −1
c
2
, c
5
y
11
− 9y
10
+ ··· + 5y −1
c
3
, c
7
y
11
− 14y
10
+ ··· + 21y −1
c
4
, c
9
y
11
− 8y
10
+ ··· + 41y −1
c
6
, c
10
y
11
− 7y
10
+ ··· + 6y −1
c
8
, c
11
y
11
+ 7y
10
+ ··· + 3y −1
c
12
y
11
− 21y
10
+ ··· + 10821y −841
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.636123 + 0.670136I
a = 1.114440 − 0.543604I
b = −0.780678 + 0.831560I
1.01561 − 3.68794I −9.46045 + 4.33638I
u = 0.636123 − 0.670136I
a = 1.114440 + 0.543604I
b = −0.780678 − 0.831560I
1.01561 + 3.68794I −9.46045 − 4.33638I
u = 0.579598 + 0.909451I
a = −0.53978 + 1.76442I
b = −1.78870 − 1.78170I
−8.26462 + 1.07707I −11.60559 + 3.04504I
u = 0.579598 − 0.909451I
a = −0.53978 − 1.76442I
b = −1.78870 + 1.78170I
−8.26462 − 1.07707I −11.60559 − 3.04504I
u = 1.16659
a = −0.621653
b = −1.35028
−8.04334 −20.3040
u = −0.693012 + 0.407011I
a = 0.911623 − 0.220211I
b = −2.23098 − 0.93364I
−1.97384 + 6.11246I −15.4629 − 13.1582I
u = −0.693012 − 0.407011I
a = 0.911623 + 0.220211I
b = −2.23098 + 0.93364I
−1.97384 − 6.11246I −15.4629 + 13.1582I
u = −0.733849
a = −1.28586
b = 0.269849
−2.65454 −14.9220
u = 0.710299
a = −0.858009
b = 2.21118
−6.08165 −31.1290
u = −1.59423 + 0.15020I
a = −0.103519 + 0.553154I
b = 0.73498 + 1.21716I
−5.41647 − 2.99510I −12.79336 + 4.15050I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.59423 − 0.15020I
a = −0.103519 − 0.553154I
b = 0.73498 − 1.21716I
−5.41647 + 2.99510I −12.79336 − 4.15050I
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
11
− 9u
10
+ ··· + 5u − 1)(u
16
− 22u
15
+ ··· + 28u + 1)
c
2
(u
11
+ 3u
10
− 6u
8
− 2u
7
+ 2u
6
+ 3u
5
+ 4u
4
+ 2u
3
− 2u
2
− 3u − 1)
· (u
16
+ 4u
15
+ ··· − 2u − 1)
c
3
(u
11
+ 2u
10
+ ··· − 5u + 1)(u
16
− 14u
15
+ ··· + 152u
2
− 32)
c
4
(u
11
− 4u
9
− 6u
8
+ 10u
7
+ 23u
6
− 10u
5
− 23u
4
− 2u
3
+ 16u
2
− 3u − 1)
· (u
16
+ u
15
+ ··· − 184u − 85)
c
5
(u
11
− 3u
10
+ 6u
8
− 2u
7
− 2u
6
+ 3u
5
− 4u
4
+ 2u
3
+ 2u
2
− 3u + 1)
· (u
16
+ 4u
15
+ ··· − 2u − 1)
c
6
(u
11
− u
10
− 3u
9
+ 5u
7
+ 4u
6
− 4u
5
− 4u
4
+ u
3
+ 3u
2
− 1)
· (u
16
− 2u
15
+ ··· − u − 1)
c
7
(u
11
− 2u
10
+ ··· − 5u − 1)(u
16
− 14u
15
+ ··· + 152u
2
− 32)
c
8
(u
11
− 3u
10
+ ··· − u + 1)(u
16
− 2u
15
+ ··· + 2u + 1)
c
9
(u
11
− 4u
9
+ 6u
8
+ 10u
7
− 23u
6
− 10u
5
+ 23u
4
− 2u
3
− 16u
2
− 3u + 1)
· (u
16
+ u
15
+ ··· − 184u − 85)
c
10
(u
11
+ u
10
− 3u
9
+ 5u
7
− 4u
6
− 4u
5
+ 4u
4
+ u
3
− 3u
2
+ 1)
· (u
16
− 2u
15
+ ··· − u − 1)
c
11
(u
11
+ 3u
10
+ ··· − u − 1)(u
16
− 2u
15
+ ··· + 2u + 1)
c
12
(u
11
− 7u
10
+ ··· − 85u + 29)(u
16
+ 8u
15
+ ··· − 17328u − 2417)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
11
− 17y
10
+ ··· + 9y −1)(y
16
+ 14y
15
+ ··· − 540y + 1)
c
2
, c
5
(y
11
− 9y
10
+ ··· + 5y −1)(y
16
+ 22y
15
+ ··· − 28y + 1)
c
3
, c
7
(y
11
− 14y
10
+ ··· + 21y −1)(y
16
− 14y
15
+ ··· − 9728y + 1024)
c
4
, c
9
(y
11
− 8y
10
+ ··· + 41y −1)(y
16
+ 15y
15
+ ··· − 36576y + 7225)
c
6
, c
10
(y
11
− 7y
10
+ ··· + 6y −1)(y
16
+ 20y
15
+ ··· − 9y + 1)
c
8
, c
11
(y
11
+ 7y
10
+ ··· + 3y −1)(y
16
+ 10y
15
+ ··· − 14y + 1)
c
12
(y
11
− 21y
10
+ ··· + 10821y −841)
· (y
16
+ 38y
15
+ ··· − 113705856y + 5841889)
15
