12n
0426
(K12n
0426
)
A knot diagram
1
Linearized knot diagam
3 6 8 12 2 9 5 12 6 8 4 10
Solving Sequence
8,12 5,9
4 3 7 6 2 11 10 1
c
8
c
4
c
3
c
7
c
6
c
2
c
11
c
10
c
12
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
6
+ 2u
5
− u
4
− u
3
− u
2
+ 3b + u − 1, 5u
6
− 16u
5
+ 11u
4
+ 20u
3
− 19u
2
+ 6a − 14u + 14,
u
7
− 4u
6
+ 5u
5
+ 2u
4
− 7u
3
+ 6u − 2i
I
u
2
= h−u
5
− 2u
4
+ u
3
+ 3u
2
+ b + u − 1, 3u
5
+ 4u
4
− 10u
3
− 11u
2
+ 2a + 5u + 12,
u
6
+ 2u
5
− 2u
4
− 5u
3
− u
2
+ 4u + 2i
I
u
3
= hb + u, a + u, u
2
+ u − 1i
I
u
4
= hb − a − u − 1, a
2
+ au + 2a + 2u + 1, u
2
+ u − 1i
I
u
5
= hu
3
− 2u
2
+ b + 2u − 1, −u
3
+ 2u
2
+ a − 3u + 2, u
4
− 2u
3
+ 4u
2
− 3u + 1i
* 5 irreducible components of dim
C
= 0, with total 23 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−u
6
+ 2u
5
− u
4
− u
3
− u
2
+ 3b + u − 1, 5u
6
− 16u
5
+ · · · + 6a +
14, u
7
− 4u
6
+ 5u
5
+ 2u
4
− 7u
3
+ 6u − 2i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
5
=
−
5
6
u
6
+
8
3
u
5
+ ··· +
7
3
u −
7
3
1
3
u
6
−
2
3
u
5
+ ··· −
1
3
u +
1
3
a
9
=
1
u
2
a
4
=
−
5
6
u
6
+
8
3
u
5
+ ··· +
7
3
u −
7
3
2
3
u
6
−
7
3
u
5
+ ··· −
8
3
u +
5
3
a
3
=
−
1
6
u
6
+
1
3
u
5
+ ··· −
1
3
u −
2
3
2
3
u
6
−
7
3
u
5
+ ··· −
8
3
u +
5
3
a
7
=
1
6
u
6
−
1
3
u
5
+ ··· −
2
3
u +
5
3
1
3
u
6
−
2
3
u
5
+ ··· −
4
3
u +
1
3
a
6
=
−
1
6
u
6
+
1
3
u
5
+ ··· −
1
3
u +
4
3
−u
6
+ 3u
5
− 2u
4
− 3u
3
+ 2u
2
+ 2u − 1
a
2
=
−
1
2
u
6
+ u
5
+
1
2
u
4
− 3u
3
+
1
2
u
2
+ 2u − 1
−
1
3
u
6
+
2
3
u
5
+ ··· +
7
3
u −
1
3
a
11
=
−
1
6
u
6
+
1
3
u
5
+ ··· +
2
3
u +
1
3
1
3
u
6
−
2
3
u
5
+ ··· −
1
3
u +
1
3
a
10
=
1
6
u
6
−
1
3
u
5
+ ··· +
1
3
u +
2
3
1
3
u
6
−
2
3
u
5
+ ··· −
1
3
u +
1
3
a
1
=
1
6
u
6
−
1
3
u
5
+ ··· +
1
3
u −
1
3
u
5
− u
4
− 2u
3
+ 2u
2
+ 3u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
6
− 6u
5
+ 4u
4
+ 6u
3
− 10u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
7
+ 6u
6
+ 27u
5
+ 62u
4
+ 93u
3
+ 76u
2
+ 36u + 4
c
2
, c
5
, c
8
u
7
+ 4u
6
+ 5u
5
− 2u
4
− 7u
3
+ 6u + 2
c
3
, c
4
, c
11
u
7
+ 3u
6
− 2u
5
− 8u
4
+ 2u
3
+ 4u
2
+ 3u + 1
c
6
, c
7
, c
9
c
12
u
7
− u
6
+ 6u
5
+ 6u
4
+ 12u
3
+ 8u
2
+ 5u + 1
c
10
u
7
− 9u
6
+ 29u
5
− 47u
4
+ 60u
3
− 40u
2
+ 12u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
7
+ 18y
6
+ 171y
5
+ 338y
4
+ 1121y
3
+ 424y
2
+ 688y −16
c
2
, c
5
, c
8
y
7
− 6y
6
+ 27y
5
− 62y
4
+ 93y
3
− 76y
2
+ 36y −4
c
3
, c
4
, c
11
y
7
− 13y
6
+ 56y
5
− 90y
4
+ 50y
3
+ 12y
2
+ y −1
c
6
, c
7
, c
9
c
12
y
7
+ 11y
6
+ 72y
5
+ 134y
4
+ 110y
3
+ 44y
2
+ 9y −1
c
10
y
7
− 23y
6
+ 115y
5
+ 575y
4
+ 608y
3
+ 216y
2
+ 464y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877051 + 0.401438I
a = 0.390423 − 0.367676I
b = 0.173321 − 0.977693I
1.59409 + 3.78166I −7.32325 − 7.33619I
u = −0.877051 − 0.401438I
a = 0.390423 + 0.367676I
b = 0.173321 + 0.977693I
1.59409 − 3.78166I −7.32325 + 7.33619I
u = 1.140270 + 0.557068I
a = 0.742429 − 0.652700I
b = 0.353960 + 0.627763I
−2.78671 − 4.23450I −16.3139 + 4.7703I
u = 1.140270 − 0.557068I
a = 0.742429 + 0.652700I
b = 0.353960 − 0.627763I
−2.78671 + 4.23450I −16.3139 − 4.7703I
u = 0.389062
a = −1.16362
b = 0.276584
−0.727542 −13.4920
u = 1.54225 + 1.02576I
a = −1.051040 + 0.651247I
b = −1.16557 − 2.38792I
4.02380 − 9.18258I −12.61674 + 3.92434I
u = 1.54225 − 1.02576I
a = −1.051040 − 0.651247I
b = −1.16557 + 2.38792I
4.02380 + 9.18258I −12.61674 − 3.92434I
5
II. I
u
2
= h−u
5
− 2u
4
+ u
3
+ 3u
2
+ b + u − 1, 3u
5
+ 4u
4
− 10u
3
− 11u
2
+ 2a +
5u + 12, u
6
+ 2u
5
− 2u
4
− 5u
3
− u
2
+ 4u + 2i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
5
=
−
3
2
u
5
− 2u
4
+ 5u
3
+
11
2
u
2
−
5
2
u − 6
u
5
+ 2u
4
− u
3
− 3u
2
− u + 1
a
9
=
1
u
2
a
4
=
−
3
2
u
5
− 2u
4
+ 5u
3
+
11
2
u
2
−
5
2
u − 6
u
5
+ 2u
4
− 2u
3
− 4u
2
+ 3
a
3
=
−
1
2
u
5
+ 3u
3
+
3
2
u
2
−
5
2
u − 3
u
5
+ 2u
4
− 2u
3
− 4u
2
+ 3
a
7
=
3
2
u
5
+ 2u
4
− 4u
3
−
9
2
u
2
+
3
2
u + 6
u
3
+ u
2
− 1
a
6
=
1
2
u
5
+ u
4
− u
3
−
5
2
u
2
+
1
2
u + 3
−u
5
− u
4
+ 4u
3
+ 3u
2
− 2u − 3
a
2
=
3
2
u
5
+ 2u
4
− 4u
3
−
7
2
u
2
+
3
2
u + 4
u
2
+ u − 1
a
11
=
−
7
2
u
5
− 5u
4
+ 10u
3
+
23
2
u
2
−
7
2
u − 12
2u
5
+ 3u
4
− 6u
3
− 7u
2
+ 3u + 7
a
10
=
−
3
2
u
5
− 2u
4
+ 4u
3
+
9
2
u
2
−
1
2
u − 5
2u
5
+ 3u
4
− 6u
3
− 7u
2
+ 3u + 7
a
1
=
9
2
u
5
+ 6u
4
− 13u
3
−
27
2
u
2
+
9
2
u + 14
−4u
5
− 6u
4
+ 11u
3
+ 15u
2
− 3u − 15
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
4
− 2u
3
+ u
2
+ 2u − 10
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
− 8u
5
+ 22u
4
− 33u
3
+ 33u
2
− 20u + 4
c
2
, c
8
u
6
+ 2u
5
− 2u
4
− 5u
3
− u
2
+ 4u + 2
c
3
, c
11
(u
3
+ 2u
2
+ 1)
2
c
4
(u
3
− 2u
2
− 1)
2
c
5
u
6
− 2u
5
− 2u
4
+ 5u
3
− u
2
− 4u + 2
c
6
u
6
− 3u
5
+ 3u
4
− 3u
3
− 3u
2
+ 4u − 1
c
7
, c
9
, c
12
u
6
+ 3u
5
+ 3u
4
+ 3u
3
− 3u
2
− 4u − 1
c
10
u
6
+ 6u
5
+ 3u
4
− 21u
3
− 5u
2
+ 25u + 13
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
− 20y
5
+ 22y
4
+ 51y
3
− 55y
2
− 136y + 16
c
2
, c
5
, c
8
y
6
− 8y
5
+ 22y
4
− 33y
3
+ 33y
2
− 20y + 4
c
3
, c
4
, c
11
(y
3
− 4y
2
− 4y −1)
2
c
6
, c
7
, c
9
c
12
y
6
− 3y
5
− 15y
4
− 5y
3
+ 27y
2
− 10y + 1
c
10
y
6
− 30y
5
+ 251y
4
− 745y
3
+ 1153y
2
− 755y + 169
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.853859 + 0.662904I
a = 0.452623 − 0.427953I
b = −0.456155 + 0.029114I
1.03690 + 2.56897I −11.22670 − 1.46771I
u = −0.853859 − 0.662904I
a = 0.452623 + 0.427953I
b = −0.456155 − 0.029114I
1.03690 − 2.56897I −11.22670 + 1.46771I
u = 1.183340 + 0.139351I
a = −0.020355 + 0.564750I
b = −0.31715 + 1.43860I
1.03690 + 2.56897I −11.22670 − 1.46771I
u = 1.183340 − 0.139351I
a = −0.020355 − 0.564750I
b = −0.31715 − 1.43860I
1.03690 − 2.56897I −11.22670 + 1.46771I
u = −0.579846
a = −3.80372
b = 0.926680
−13.5883 −10.5470
u = −2.07912
a = −1.06082
b = −2.38008
−13.5883 −10.5470
9
III. I
u
3
= hb + u, a + u, u
2
+ u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
5
=
−u
−u
a
9
=
1
−u + 1
a
4
=
−u
u − 1
a
3
=
−1
u − 1
a
7
=
u
u − 1
a
6
=
0
−u
a
2
=
−1
0
a
11
=
2u − 1
−2u + 2
a
10
=
1
−2u + 2
a
1
=
−u
−3u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −20
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
11
u
2
− 3u + 1
c
2
, c
6
, c
8
u
2
+ u − 1
c
4
u
2
+ 3u + 1
c
5
, c
7
, c
9
c
12
u
2
− u − 1
c
10
u
2
+ 6u + 4
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
11
y
2
− 7y + 1
c
2
, c
5
, c
6
c
7
, c
8
, c
9
c
12
y
2
− 3y + 1
c
10
y
2
− 28y + 16
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −0.618034
b = −0.618034
−1.97392 −20.0000
u = −1.61803
a = 1.61803
b = 1.61803
−17.7653 −20.0000
13
IV. I
u
4
= hb − a − u − 1, a
2
+ au + 2a + 2u + 1, u
2
+ u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
5
=
a
a + u + 1
a
9
=
1
−u + 1
a
4
=
a
au + u + 1
a
3
=
au + a + u + 1
au + u + 1
a
7
=
a + 2u + 2
−au + u − 1
a
6
=
u + 1
−a − 1
a
2
=
au + a + u + 2
1
a
11
=
−au − a + u − 2
−2u + 1
a
10
=
−au − a − u − 1
−2u + 1
a
1
=
a + 2
−2au + a − u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −14
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
+ 3u + 1)
2
c
2
, c
5
, c
8
(u
2
− u − 1)
2
c
3
, c
4
, c
11
u
4
+ u
3
− 6u
2
− 10u − 5
c
6
, c
7
, c
9
c
12
u
4
− u
3
− 2u
2
− 2u − 1
c
10
(u
2
+ 4u − 1)
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
2
− 7y + 1)
2
c
2
, c
5
, c
8
(y
2
− 3y + 1)
2
c
3
, c
4
, c
11
y
4
− 13y
3
+ 46y
2
− 40y + 25
c
6
, c
7
, c
9
c
12
y
4
− 5y
3
− 2y
2
+ 1
c
10
(y
2
− 18y + 1)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −1.30902 + 0.72287I
b = 0.309017 + 0.722871I
−0.328987 −14.0000
u = 0.618034
a = −1.30902 − 0.72287I
b = 0.309017 − 0.722871I
−0.328987 −14.0000
u = −1.61803
a = 1.31651
b = 0.698478
−16.1204 −14.0000
u = −1.61803
a = −1.69848
b = −2.31651
−16.1204 −14.0000
17
V.
I
u
5
= hu
3
− 2u
2
+ b + 2u − 1, −u
3
+ 2u
2
+ a − 3u + 2, u
4
− 2u
3
+ 4u
2
− 3u + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
5
=
u
3
− 2u
2
+ 3u − 2
−u
3
+ 2u
2
− 2u + 1
a
9
=
1
u
2
a
4
=
u
3
− 2u
2
+ 3u − 2
u
2
− u + 1
a
3
=
u
3
− u
2
+ 2u − 1
u
2
− u + 1
a
7
=
u
2
− 2u + 2
−u
3
− u
2
+ 2u − 1
a
6
=
−u
3
+ 2u
2
− 3u + 2
0
a
2
=
0
u
2
− u + 1
a
11
=
−u + 1
u
2
a
10
=
u
2
− u + 1
u
2
a
1
=
u
3
− u
2
u
3
+ u
2
− u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
2
+ u − 13
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
4
− 4u
3
+ 6u
2
+ u + 1
c
2
, c
5
, c
8
u
4
+ 2u
3
+ 4u
2
+ 3u + 1
c
3
, c
4
, c
11
(u
2
− u − 1)
2
c
6
, c
7
, c
9
c
12
u
4
− u
3
+ 6u
2
+ 4u + 1
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
4
− 4y
3
+ 46y
2
+ 11y + 1
c
2
, c
5
, c
8
y
4
+ 4y
3
+ 6y
2
− y + 1
c
3
, c
4
, c
11
(y
2
− 3y + 1)
2
c
6
, c
7
, c
9
c
12
y
4
+ 11y
3
+ 46y
2
− 4y + 1
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.363271I
a = −0.809017 + 0.587785I
b = 0.309017 − 0.224514I
−0.657974 −12.61803 + 0.I
u = 0.500000 − 0.363271I
a = −0.809017 − 0.587785I
b = 0.309017 + 0.224514I
−0.657974 −12.61803 + 0.I
u = 0.50000 + 1.53884I
a = 0.309017 − 0.951057I
b = −0.80902 + 2.48990I
7.23771 −10.38197 + 0.I
u = 0.50000 − 1.53884I
a = 0.309017 + 0.951057I
b = −0.80902 − 2.48990I
7.23771 −10.38197 + 0.I
21
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− 3u + 1)(u
2
+ 3u + 1)
2
(u
4
− 4u
3
+ 6u
2
+ u + 1)
· (u
6
− 8u
5
+ 22u
4
− 33u
3
+ 33u
2
− 20u + 4)
· (u
7
+ 6u
6
+ 27u
5
+ 62u
4
+ 93u
3
+ 76u
2
+ 36u + 4)
c
2
, c
8
(u
2
− u − 1)
2
(u
2
+ u − 1)(u
4
+ 2u
3
+ 4u
2
+ 3u + 1)
· (u
6
+ 2u
5
− 2u
4
− 5u
3
− u
2
+ 4u + 2)
· (u
7
+ 4u
6
+ 5u
5
− 2u
4
− 7u
3
+ 6u + 2)
c
3
, c
11
(u
2
− 3u + 1)(u
2
− u − 1)
2
(u
3
+ 2u
2
+ 1)
2
(u
4
+ u
3
+ ··· − 10u − 5)
· (u
7
+ 3u
6
− 2u
5
− 8u
4
+ 2u
3
+ 4u
2
+ 3u + 1)
c
4
((u
2
− u − 1)
2
)(u
2
+ 3u + 1)(u
3
− 2u
2
− 1)
2
(u
4
+ u
3
+ ··· − 10u − 5)
· (u
7
+ 3u
6
− 2u
5
− 8u
4
+ 2u
3
+ 4u
2
+ 3u + 1)
c
5
(u
2
− u − 1)
3
(u
4
+ 2u
3
+ 4u
2
+ 3u + 1)
· (u
6
− 2u
5
− 2u
4
+ 5u
3
− u
2
− 4u + 2)
· (u
7
+ 4u
6
+ 5u
5
− 2u
4
− 7u
3
+ 6u + 2)
c
6
(u
2
+ u − 1)(u
4
− u
3
− 2u
2
− 2u − 1)(u
4
− u
3
+ 6u
2
+ 4u + 1)
· (u
6
− 3u
5
+ 3u
4
− 3u
3
− 3u
2
+ 4u − 1)
· (u
7
− u
6
+ 6u
5
+ 6u
4
+ 12u
3
+ 8u
2
+ 5u + 1)
c
7
, c
9
, c
12
(u
2
− u − 1)(u
4
− u
3
− 2u
2
− 2u − 1)(u
4
− u
3
+ 6u
2
+ 4u + 1)
· (u
6
+ 3u
5
+ 3u
4
+ 3u
3
− 3u
2
− 4u − 1)
· (u
7
− u
6
+ 6u
5
+ 6u
4
+ 12u
3
+ 8u
2
+ 5u + 1)
c
10
(u
2
+ 4u − 1)
2
(u
2
+ 6u + 4)(u
4
− 4u
3
+ 6u
2
+ u + 1)
· (u
6
+ 6u
5
+ 3u
4
− 21u
3
− 5u
2
+ 25u + 13)
· (u
7
− 9u
6
+ 29u
5
− 47u
4
+ 60u
3
− 40u
2
+ 12u + 4)
22
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
− 7y + 1)
3
(y
4
− 4y
3
+ 46y
2
+ 11y + 1)
· (y
6
− 20y
5
+ 22y
4
+ 51y
3
− 55y
2
− 136y + 16)
· (y
7
+ 18y
6
+ 171y
5
+ 338y
4
+ 1121y
3
+ 424y
2
+ 688y −16)
c
2
, c
5
, c
8
(y
2
− 3y + 1)
3
(y
4
+ 4y
3
+ 6y
2
− y + 1)
· (y
6
− 8y
5
+ 22y
4
− 33y
3
+ 33y
2
− 20y + 4)
· (y
7
− 6y
6
+ 27y
5
− 62y
4
+ 93y
3
− 76y
2
+ 36y −4)
c
3
, c
4
, c
11
(y
2
− 7y + 1)(y
2
− 3y + 1)
2
(y
3
− 4y
2
− 4y −1)
2
· (y
4
− 13y
3
+ 46y
2
− 40y + 25)
· (y
7
− 13y
6
+ 56y
5
− 90y
4
+ 50y
3
+ 12y
2
+ y −1)
c
6
, c
7
, c
9
c
12
(y
2
− 3y + 1)(y
4
− 5y
3
− 2y
2
+ 1)(y
4
+ 11y
3
+ 46y
2
− 4y + 1)
· (y
6
− 3y
5
− 15y
4
− 5y
3
+ 27y
2
− 10y + 1)
· (y
7
+ 11y
6
+ 72y
5
+ 134y
4
+ 110y
3
+ 44y
2
+ 9y −1)
c
10
(y
2
− 28y + 16)(y
2
− 18y + 1)
2
(y
4
− 4y
3
+ 46y
2
+ 11y + 1)
· (y
6
− 30y
5
+ 251y
4
− 745y
3
+ 1153y
2
− 755y + 169)
· (y
7
− 23y
6
+ 115y
5
+ 575y
4
+ 608y
3
+ 216y
2
+ 464y −16)
23
