12n
0328
(K12n
0328
)
A knot diagram
1
Linearized knot diagam
3 6 8 10 2 4 10 12 4 6 8 11
Solving Sequence
4,8 3,10
7 6 11 12 2 1 5 9
c
3
c
7
c
6
c
10
c
11
c
2
c
1
c
5
c
9
c
4
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−4283194u
11
+ 2308461u
10
+ ··· + 21359143b − 4337843,
− 139682999u
11
+ 35486115u
10
+ ··· + 21359143a − 526213370,
u
12
+ u
10
+ 10u
9
+ u
8
− 6u
7
− 28u
6
− 45u
5
− 26u
4
− 34u
3
− 3u
2
+ 5u + 1i
I
u
2
= h−u
5
− 2u
4
− u
3
− 5u
2
+ 2b + 1, −u
5
+ u
4
+ 4u
3
− 2u
2
+ 4a + 13u − 3, u
6
+ 2u
5
+ u
4
+ 4u
3
− u
2
− 2u − 1i
I
u
3
= hu
2
+ b, −u
2
+ a + u + 1, u
3
− u
2
− 2u + 1i
I
u
4
= hu
3
+ u
2
+ 6b + 3u + 1, −2u
3
− 23u
2
+ 138a + 30u − 77, u
4
+ 8u
2
+ 4u + 23i
I
u
5
= hb − 1, a
2
+ a + 1, u − 1i
* 5 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−4.28 × 10
6
u
11
+ 2.31 × 10
6
u
10
+ · · · + 2.14 × 10
7
b − 4.34 × 10
6
, −1.40 ×
10
8
u
11
+ 3.55× 10
7
u
10
+ · · · + 2.14 × 10
7
a −5.26 × 10
8
, u
12
+ u
10
+ · · · + 5u + 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
3
=
1
−u
2
a
10
=
6.53973u
11
− 1.66140u
10
+ ··· + 29.8803u + 24.6364
0.200532u
11
− 0.108078u
10
+ ··· + 3.23080u + 0.203091
a
7
=
−9.40115u
11
+ 2.69041u
10
+ ··· − 50.5716u − 32.5778
1.66140u
11
− 0.430541u
10
+ ··· + 8.06220u + 6.53973
a
6
=
−7.73975u
11
+ 2.25987u
10
+ ··· − 42.5094u − 26.0381
1.66140u
11
− 0.430541u
10
+ ··· + 8.06220u + 6.53973
a
11
=
4.27986u
11
− 1.02645u
10
+ ··· + 17.2197u + 16.8967
0.631073u
11
− 0.239539u
10
+ ··· + 4.99808u + 1.86449
a
12
=
4.27986u
11
− 1.02645u
10
+ ··· + 17.2197u + 16.8967
0.874490u
11
− 0.282949u
10
+ ··· + 5.85047u + 2.89094
a
2
=
4.87432u
11
− 1.53943u
10
+ ··· + 31.4054u + 15.7081
−2.28293u
11
+ 0.628514u
10
+ ··· − 11.6846u − 8.27969
a
1
=
3.10484u
11
− 1.07242u
10
+ ··· + 22.5436u + 8.96784
−2.13788u
11
+ 0.599560u
10
+ ··· − 11.1190u − 7.81268
a
5
=
−6.53973u
11
+ 1.66140u
10
+ ··· − 29.8803u − 24.6364
−0.641879u
11
+ 0.236936u
10
+ ··· − 4.78908u − 1.44344
a
9
=
6.74026u
11
− 1.76948u
10
+ ··· + 33.1111u + 24.8395
0.200532u
11
− 0.108078u
10
+ ··· + 3.23080u + 0.203091
(ii) Obstruction class = −1
(iii) Cusp Shapes =
16757967
1643011
u
11
−
5020278
1643011
u
10
+ ··· +
1299731
23141
u +
33667882
1643011
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
12
u
12
+ 11u
11
+ ··· + 22u + 1
c
2
, c
5
, c
8
c
11
u
12
+ u
11
+ ··· − 2u − 1
c
3
, c
10
u
12
+ u
10
+ ··· − 5u + 1
c
4
, c
9
u
12
− 4u
11
+ ··· + 3u + 1
c
6
, c
7
u
12
− u
11
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
12
y
12
− 11y
11
+ ··· − 154y + 1
c
2
, c
5
, c
8
c
11
y
12
− 11y
11
+ ··· − 22y + 1
c
3
, c
10
y
12
+ 2y
11
+ ··· − 31y + 1
c
4
, c
9
y
12
− 30y
11
+ ··· + 45y + 1
c
6
, c
7
y
12
− 7y
11
+ ··· − 14y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.141786 + 0.980425I
a = −0.933540 + 0.158952I
b = 0.041012 + 0.177252I
1.74171 − 4.08194I −5.82114 + 7.56540I
u = 0.141786 − 0.980425I
a = −0.933540 − 0.158952I
b = 0.041012 − 0.177252I
1.74171 + 4.08194I −5.82114 − 7.56540I
u = −0.442926 + 1.140420I
a = −0.476761 + 0.082105I
b = 1.037070 + 0.350813I
−1.35590 + 2.26651I −17.0173 − 3.2135I
u = −0.442926 − 1.140420I
a = −0.476761 − 0.082105I
b = 1.037070 − 0.350813I
−1.35590 − 2.26651I −17.0173 + 3.2135I
u = −1.29341
a = 1.70253
b = −1.58736
−7.73551 −2.14220
u = 0.362550
a = −0.697529
b = 0.433632
−0.619674 −15.7990
u = 1.68235
a = 1.15411
b = −1.86647
−12.8318 −20.4330
u = −0.277013 + 0.027736I
a = 2.61966 + 3.59517I
b = −1.132390 + 0.181685I
−1.88495 − 4.13308I −16.9007 + 5.9544I
u = −0.277013 − 0.027736I
a = 2.61966 − 3.59517I
b = −1.132390 − 0.181685I
−1.88495 + 4.13308I −16.9007 − 5.9544I
u = −1.90708
a = −0.231901
b = 3.31219
−18.8402 −5.38040
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.15595 + 2.12191I
a = 0.827033 − 0.271266I
b = −2.09169 − 0.35807I
4.24091 − 10.44050I −12.38352 + 4.77186I
u = 1.15595 − 2.12191I
a = 0.827033 + 0.271266I
b = −2.09169 + 0.35807I
4.24091 + 10.44050I −12.38352 − 4.77186I
6
II. I
u
2
= h−u
5
− 2u
4
− u
3
− 5u
2
+ 2b + 1, −u
5
+ u
4
+ 4u
3
− 2u
2
+ 4a + 13u −
3, u
6
+ 2u
5
+ u
4
+ 4u
3
− u
2
− 2u − 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
3
=
1
−u
2
a
10
=
1
4
u
5
−
1
4
u
4
+ ··· −
13
4
u +
3
4
1
2
u
5
+ u
4
+
1
2
u
3
+
5
2
u
2
−
1
2
a
7
=
−
5
4
u
5
−
7
4
u
4
+ ··· +
19
4
u +
11
4
−
1
4
u
5
−
3
4
u
4
+ ··· −
3
4
u +
1
4
a
6
=
−
3
2
u
5
−
5
2
u
4
+ ··· + 4u + 3
−
1
4
u
5
−
3
4
u
4
+ ··· −
3
4
u +
1
4
a
11
=
1
4
u
5
+
3
4
u
4
+ ··· +
5
4
u −
7
4
−
1
4
u
5
−
3
4
u
4
+ ··· −
3
4
u −
3
4
a
12
=
1
4
u
5
+
3
4
u
4
+ ··· +
5
4
u −
7
4
−
1
2
u
5
− u
4
−
1
2
u
3
−
5
2
u
2
−
1
2
a
2
=
2u
5
+
7
2
u
4
+ u
3
+ 8u
2
− 3u −
5
2
−
1
4
u
5
−
3
4
u
4
+ ··· −
3
4
u −
3
4
a
1
=
7
4
u
5
+
13
4
u
4
+ ··· −
11
4
u −
15
4
−u
2
− u − 1
a
5
=
1
4
u
5
+
1
4
u
4
+ ··· −
7
4
u −
5
4
1
2
u
4
+ u
3
+ u
2
+ 2u +
1
2
a
9
=
3
4
u
5
+
3
4
u
4
+ ··· −
13
4
u +
1
4
1
2
u
5
+ u
4
+
1
2
u
3
+
5
2
u
2
−
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
5
+
3
2
u
4
+ 4u
2
− 2u −
25
2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
− 5u
5
+ 10u
4
− 13u
3
+ 14u
2
− 9u + 1
c
2
, c
8
u
6
+ u
5
− 2u
4
− 3u
3
+ 3u + 1
c
3
u
6
+ 2u
5
+ u
4
+ 4u
3
− u
2
− 2u − 1
c
4
(u
3
− u
2
− u − 1)
2
c
5
, c
11
u
6
− u
5
− 2u
4
+ 3u
3
− 3u + 1
c
6
u
6
+ u
5
− 2u
4
− 5u
3
− 8u
2
− 5u − 1
c
7
u
6
− u
5
− 2u
4
+ 5u
3
− 8u
2
+ 5u − 1
c
9
(u
3
+ u
2
− u + 1)
2
c
10
u
6
− 2u
5
+ u
4
− 4u
3
− u
2
+ 2u − 1
c
12
u
6
+ 5u
5
+ 10u
4
+ 13u
3
+ 14u
2
+ 9u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
12
y
6
− 5y
5
− 2y
4
+ 23y
3
− 18y
2
− 53y + 1
c
2
, c
5
, c
8
c
11
y
6
− 5y
5
+ 10y
4
− 13y
3
+ 14y
2
− 9y + 1
c
3
, c
10
y
6
− 2y
5
− 17y
4
− 12y
3
+ 15y
2
− 2y + 1
c
4
, c
9
(y
3
− 3y
2
− y −1)
2
c
6
, c
7
y
6
− 5y
5
− 2y
4
+ 15y
3
+ 18y
2
− 9y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.788614
a = −2.01293
b = 1.83929
−11.8065 −10.7040
u = 0.15540 + 1.44647I
a = −0.005444 + 0.311582I
b = −0.419643 + 0.606291I
0.96847 − 3.17729I −11.64780 + 1.72143I
u = 0.15540 − 1.44647I
a = −0.005444 − 0.311582I
b = −0.419643 − 0.606291I
0.96847 + 3.17729I −11.64780 − 1.72143I
u = −0.383557 + 0.331324I
a = 1.96878 − 1.31241I
b = −0.419643 − 0.606291I
0.96847 + 3.17729I −11.64780 − 1.72143I
u = −0.383557 − 0.331324I
a = 1.96878 + 1.31241I
b = −0.419643 + 0.606291I
0.96847 − 3.17729I −11.64780 + 1.72143I
u = −2.33230
a = −0.913737
b = 1.83929
−11.8065 −10.7040
10
III. I
u
3
= hu
2
+ b, −u
2
+ a + u + 1, u
3
− u
2
− 2u + 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
3
=
1
−u
2
a
10
=
u
2
− u − 1
−u
2
a
7
=
−u
2
+ 2u
−u + 1
a
6
=
−u
2
+ u + 1
−u + 1
a
11
=
u
2
− 2
−u
a
12
=
u
2
− 2
u
2
− 1
a
2
=
0
−u
2
− u
a
1
=
−u
2
− u
3u
2
+ 2u − 2
a
5
=
−u
2
+ u + 1
3u
2
+ u − 1
a
9
=
−u − 1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5u
2
+ 6u − 19
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
− 6u
2
+ 5u − 1
c
2
, c
6
, c
8
u
3
+ 2u
2
− u − 1
c
3
u
3
− u
2
− 2u + 1
c
4
u
3
+ 5u
2
+ 6u + 1
c
5
, c
7
, c
11
u
3
− 2u
2
− u + 1
c
9
u
3
− 5u
2
+ 6u − 1
c
10
u
3
+ u
2
− 2u − 1
c
12
u
3
+ 6u
2
+ 5u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
12
y
3
− 26y
2
+ 13y −1
c
2
, c
5
, c
6
c
7
, c
8
, c
11
y
3
− 6y
2
+ 5y −1
c
3
, c
10
y
3
− 5y
2
+ 6y −1
c
4
, c
9
y
3
− 13y
2
+ 26y −1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.24698
a = 1.80194
b = −1.55496
−7.98968 −34.2570
u = 0.445042
a = −1.24698
b = −0.198062
−2.34991 −17.3200
u = 1.80194
a = 0.445042
b = −3.24698
−19.2692 −24.4230
14
IV.
I
u
4
= hu
3
+u
2
+6b+3u+1, −2u
3
−23u
2
+138a+30u−77, u
4
+8u
2
+4u+23i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
3
=
1
−u
2
a
10
=
1
69
u
3
+
1
6
u
2
−
5
23
u +
77
138
−
1
6
u
3
−
1
6
u
2
−
1
2
u −
1
6
a
7
=
0.0507246u
3
+ 0.333333u
2
+ 0.239130u + 2.20290
−
1
2
u
2
−
5
2
a
6
=
0.0507246u
3
− 0.166667u
2
+ 0.239130u − 0.297101
−
1
2
u
2
−
5
2
a
11
=
−
3
46
u
3
−
1
46
u −
6
23
−
2
3
u
3
−
1
6
u
2
− 2u −
7
6
a
12
=
−
3
46
u
3
−
1
46
u −
6
23
−
1
6
u
3
−
1
6
u
2
−
1
2
u −
7
6
a
2
=
1
23
u
3
+
8
23
u +
4
23
−
1
2
u
2
−
1
2
a
1
=
1
23
u
3
−
1
2
u
2
−
15
23
u −
15
46
u
3
− 4u
2
− 2u − 12
a
5
=
5
46
u
3
+
17
46
u +
10
23
−
1
3
u
3
−
1
3
u
2
− u +
2
3
a
9
=
−
7
46
u
3
−
33
46
u +
9
23
−
1
6
u
3
−
1
6
u
2
−
1
2
u −
1
6
(ii) Obstruction class = −1
(iii) Cusp Shapes = −10
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
12
u
4
− 6u
3
+ 31u
2
− 66u + 49
c
2
, c
5
, c
8
c
11
u
4
+ 2u
3
+ 5u
2
+ 2u + 7
c
3
, c
10
u
4
+ 8u
2
− 4u + 23
c
4
, c
9
(u
2
+ 2u − 1)
2
c
6
, c
7
u
4
− 2u
3
+ 5u
2
− 6u + 9
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
12
y
4
+ 26y
3
+ 267y
2
− 1318y + 2401
c
2
, c
5
, c
8
c
11
y
4
+ 6y
3
+ 31y
2
+ 66y + 49
c
3
, c
10
y
4
+ 16y
3
+ 110y
2
+ 352y + 529
c
4
, c
9
(y
2
− 6y + 1)
2
c
6
, c
7
y
4
+ 6y
3
+ 19y
2
+ 54y + 81
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.70711 + 1.75664I
a = 0.370470 − 0.836294I
b = −0.414214
4.93480 −10.0000
u = −0.70711 − 1.75664I
a = 0.370470 + 0.836294I
b = −0.414214
4.93480 −10.0000
u = 0.70711 + 2.43192I
a = −0.674818 − 0.111049I
b = 2.41421
4.93480 −10.0000
u = 0.70711 − 2.43192I
a = −0.674818 + 0.111049I
b = 2.41421
4.93480 −10.0000
18
V. I
u
5
= hb − 1, a
2
+ a + 1, u − 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
1
a
3
=
1
−1
a
10
=
a
1
a
7
=
−a − 1
a + 1
a
6
=
0
a + 1
a
11
=
a
−a
a
12
=
a
0
a
2
=
1
−a − 1
a
1
=
−a + 1
−1
a
5
=
a + 1
1
a
9
=
a + 1
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −12
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
12
u
2
− u + 1
c
2
, c
5
, c
6
c
7
, c
8
, c
11
u
2
+ u + 1
c
3
, c
4
, c
9
c
10
(u + 1)
2
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
8
c
11
, c
12
y
2
+ y + 1
c
3
, c
4
, c
9
c
10
(y −1)
2
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.500000 + 0.866025I
b = 1.00000
0 −12.0000
u = 1.00000
a = −0.500000 − 0.866025I
b = 1.00000
0 −12.0000
22
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)(u
3
− 6u
2
+ 5u − 1)(u
4
− 6u
3
+ 31u
2
− 66u + 49)
· (u
6
− 5u
5
+ ··· − 9u + 1)(u
12
+ 11u
11
+ ··· + 22u + 1)
c
2
, c
8
(u
2
+ u + 1)(u
3
+ 2u
2
− u − 1)(u
4
+ 2u
3
+ 5u
2
+ 2u + 7)
· (u
6
+ u
5
− 2u
4
− 3u
3
+ 3u + 1)(u
12
+ u
11
+ ··· − 2u − 1)
c
3
(u + 1)
2
(u
3
− u
2
− 2u + 1)(u
4
+ 8u
2
− 4u + 23)
· (u
6
+ 2u
5
+ u
4
+ 4u
3
− u
2
− 2u − 1)(u
12
+ u
10
+ ··· − 5u + 1)
c
4
(u + 1)
2
(u
2
+ 2u − 1)
2
(u
3
− u
2
− u − 1)
2
(u
3
+ 5u
2
+ 6u + 1)
· (u
12
− 4u
11
+ ··· + 3u + 1)
c
5
, c
11
(u
2
+ u + 1)(u
3
− 2u
2
− u + 1)(u
4
+ 2u
3
+ 5u
2
+ 2u + 7)
· (u
6
− u
5
− 2u
4
+ 3u
3
− 3u + 1)(u
12
+ u
11
+ ··· − 2u − 1)
c
6
(u
2
+ u + 1)(u
3
+ 2u
2
− u − 1)(u
4
− 2u
3
+ 5u
2
− 6u + 9)
· (u
6
+ u
5
− 2u
4
− 5u
3
− 8u
2
− 5u − 1)(u
12
− u
11
+ ··· + 2u + 1)
c
7
(u
2
+ u + 1)(u
3
− 2u
2
− u + 1)(u
4
− 2u
3
+ 5u
2
− 6u + 9)
· (u
6
− u
5
− 2u
4
+ 5u
3
− 8u
2
+ 5u − 1)(u
12
− u
11
+ ··· + 2u + 1)
c
9
(u + 1)
2
(u
2
+ 2u − 1)
2
(u
3
− 5u
2
+ 6u − 1)(u
3
+ u
2
− u + 1)
2
· (u
12
− 4u
11
+ ··· + 3u + 1)
c
10
(u + 1)
2
(u
3
+ u
2
− 2u − 1)(u
4
+ 8u
2
− 4u + 23)
· (u
6
− 2u
5
+ u
4
− 4u
3
− u
2
+ 2u − 1)(u
12
+ u
10
+ ··· − 5u + 1)
c
12
(u
2
− u + 1)(u
3
+ 6u
2
+ 5u + 1)(u
4
− 6u
3
+ 31u
2
− 66u + 49)
· (u
6
+ 5u
5
+ ··· + 9u + 1)(u
12
+ 11u
11
+ ··· + 22u + 1)
23
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
12
(y
2
+ y + 1)(y
3
− 26y
2
+ 13y −1)(y
4
+ 26y
3
+ ··· − 1318y + 2401)
· (y
6
− 5y
5
+ ··· − 53y + 1)(y
12
− 11y
11
+ ··· − 154y + 1)
c
2
, c
5
, c
8
c
11
(y
2
+ y + 1)(y
3
− 6y
2
+ 5y −1)(y
4
+ 6y
3
+ 31y
2
+ 66y + 49)
· (y
6
− 5y
5
+ ··· − 9y + 1)(y
12
− 11y
11
+ ··· − 22y + 1)
c
3
, c
10
(y −1)
2
(y
3
− 5y
2
+ 6y −1)(y
4
+ 16y
3
+ 110y
2
+ 352y + 529)
· (y
6
− 2y
5
+ ··· − 2y + 1)(y
12
+ 2y
11
+ ··· − 31y + 1)
c
4
, c
9
(y −1)
2
(y
2
− 6y + 1)
2
(y
3
− 13y
2
+ 26y −1)(y
3
− 3y
2
− y −1)
2
· (y
12
− 30y
11
+ ··· + 45y + 1)
c
6
, c
7
(y
2
+ y + 1)(y
3
− 6y
2
+ 5y −1)(y
4
+ 6y
3
+ 19y
2
+ 54y + 81)
· (y
6
− 5y
5
+ ··· − 9y + 1)(y
12
− 7y
11
+ ··· − 14y + 1)
24
