12n
0462
(K12n
0462
)
A knot diagram
1
Linearized knot diagam
3 5 8 12 2 9 12 3 5 8 3 10
Solving Sequence
3,5
2
1,10
9 8 12 4 7 6 11
c
2
c
1
c
9
c
8
c
12
c
4
c
7
c
6
c
11
c
3
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−149788246u
19
+ 735301154u
18
+ ··· + 16882806339b − 24202510048,
− 38489542585u
19
+ 46865050127u
18
+ ··· + 16882806339a − 207132124495,
u
20
− u
19
+ ··· + 11u + 1i
I
u
2
= hu
2
+ b + 1, u
5
− 2u
4
+ 5u
3
− 6u
2
+ 3a + 6u − 1, u
6
− u
5
+ 5u
4
− 4u
3
+ 7u
2
− 2u + 3i
I
u
3
= hb − u, a − u, u
2
+ u + 1i
* 3 irreducible components of dim
C
= 0, with total 28 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−1.50×10
8
u
19
+7.35×10
8
u
18
+· · ·+1.69×10
10
b−2.42×10
10
, −3.85×
10
10
u
19
+4.69×10
10
u
18
+· · ·+1.69×10
10
a−2.07×10
11
, u
20
−u
19
+· · ·+11u+1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
1
=
u
2
+ 1
u
2
a
10
=
2.27981u
19
− 2.77590u
18
+ ··· + 74.2433u + 12.2688
0.00887224u
19
− 0.0435533u
18
+ ··· + 9.24808u + 1.43356
a
9
=
2.27981u
19
− 2.77590u
18
+ ··· + 74.2433u + 12.2688
0.0852452u
19
− 0.0369754u
18
+ ··· + 12.4253u + 1.92966
a
8
=
2.36505u
19
− 2.81288u
18
+ ··· + 86.6686u + 14.1985
0.0852452u
19
− 0.0369754u
18
+ ··· + 12.4253u + 1.92966
a
12
=
−0.00359098u
19
+ 0.391699u
18
+ ··· + 9.11164u + 4.91265
0.327637u
19
− 0.499312u
18
+ ··· + 0.549227u + 0.495867
a
4
=
0.627286u
19
− 0.803555u
18
+ ··· + 31.8391u + 2.03483
0.253178u
19
− 0.0827151u
18
+ ··· + 12.4507u + 0.748328
a
7
=
1.92966u
19
− 2.01490u
18
+ ··· + 73.2986u + 8.80089
0.509356u
19
− 0.765818u
18
+ ··· + 2.50917u + 0.308552
a
6
=
−u
−u
3
− u
a
11
=
0.324046u
19
− 0.107613u
18
+ ··· + 9.66087u + 5.40852
0.327637u
19
− 0.499312u
18
+ ··· + 0.549227u + 0.495867
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
12674828861
16882806339
u
19
+
1750449202
2411829477
u
18
+ ···−
510127682849
16882806339
u +
10680247402
16882806339
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
+ 27u
19
+ ··· − 27u + 1
c
2
, c
5
u
20
+ u
19
+ ··· − 11u + 1
c
3
, c
8
u
20
− u
19
+ ··· + 11u + 1
c
4
u
20
+ 2u
19
+ ··· − 27u + 51
c
6
u
20
+ 16u
18
+ ··· − 16u + 52
c
7
u
20
− 3u
19
+ ··· + 109u
2
+ 21
c
9
u
20
− 2u
19
+ ··· + 27u + 51
c
10
u
20
+ 5u
19
+ ··· + 57u + 7
c
11
u
20
+ 16u
18
+ ··· + 16u + 52
c
12
u
20
− 5u
19
+ ··· − 57u + 7
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
− 61y
19
+ ··· + 169y + 1
c
2
, c
3
, c
5
c
8
y
20
+ 27y
19
+ ··· − 27y + 1
c
4
, c
9
y
20
+ 18y
19
+ ··· + 17835y + 2601
c
6
, c
11
y
20
+ 32y
19
+ ··· + 7024y + 2704
c
7
y
20
− 33y
19
+ ··· + 4578y + 441
c
10
, c
12
y
20
− 15y
19
+ ··· − 1835y + 49
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.160143 + 0.768509I
a = −0.394995 − 1.004840I
b = −0.636041 − 0.396936I
−1.10947 − 1.48655I −5.12329 + 2.41841I
u = 0.160143 − 0.768509I
a = −0.394995 + 1.004840I
b = −0.636041 + 0.396936I
−1.10947 + 1.48655I −5.12329 − 2.41841I
u = −0.484926 + 0.607075I
a = 0.577525 − 0.647733I
b = −0.026523 − 0.353444I
− 1.43025I 0. + 5.92138I
u = −0.484926 − 0.607075I
a = 0.577525 + 0.647733I
b = −0.026523 + 0.353444I
1.43025I 0. − 5.92138I
u = −0.08307 + 1.42113I
a = −0.271547 + 1.012220I
b = −1.110600 + 0.490957I
3.72589 − 1.03786I 1.73919 + 0.58908I
u = −0.08307 − 1.42113I
a = −0.271547 − 1.012220I
b = −1.110600 − 0.490957I
3.72589 + 1.03786I 1.73919 − 0.58908I
u = 1.18575 + 0.83320I
a = 0.626136 + 0.573895I
b = 0.569199 + 0.092819I
8.81653 + 3.95168I 0.25331 − 3.24699I
u = 1.18575 − 0.83320I
a = 0.626136 − 0.573895I
b = 0.569199 − 0.092819I
8.81653 − 3.95168I 0.25331 + 3.24699I
u = −0.104414 + 0.507262I
a = 1.44705 − 1.63599I
b = 0.64554 + 1.51309I
10.23410 − 0.33723I 1.46943 − 0.53181I
u = −0.104414 − 0.507262I
a = 1.44705 + 1.63599I
b = 0.64554 − 1.51309I
10.23410 + 0.33723I 1.46943 + 0.53181I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.32507 + 1.48224I
a = 0.843038 − 0.018166I
b = 1.69755 − 0.64672I
−3.72589 − 1.03786I −1.73919 + 0.58908I
u = −0.32507 − 1.48224I
a = 0.843038 + 0.018166I
b = 1.69755 + 0.64672I
−3.72589 + 1.03786I −1.73919 − 0.58908I
u = −0.11912 + 1.73852I
a = −1.211980 − 0.171698I
b = −2.38905 − 0.18528I
−8.81653 − 3.95168I −0.25331 + 3.24699I
u = −0.11912 − 1.73852I
a = −1.211980 + 0.171698I
b = −2.38905 + 0.18528I
−8.81653 + 3.95168I −0.25331 − 3.24699I
u = 0.11567 + 1.76293I
a = 0.697284 + 0.000472I
b = 2.05356 + 0.22436I
−10.23410 + 0.33723I −1.46943 + 0.53181I
u = 0.11567 − 1.76293I
a = 0.697284 − 0.000472I
b = 2.05356 − 0.22436I
−10.23410 − 0.33723I −1.46943 − 0.53181I
u = 0.32058 + 1.78463I
a = −0.974674 + 0.336167I
b = −2.29883 + 0.36675I
9.75717I 0. − 4.10936I
u = 0.32058 − 1.78463I
a = −0.974674 − 0.336167I
b = −2.29883 − 0.36675I
− 9.75717I 0. + 4.10936I
u = −0.165551 + 0.073534I
a = 2.16216 + 4.82053I
b = −0.004791 + 0.715102I
1.10947 + 1.48655I 5.12329 − 2.41841I
u = −0.165551 − 0.073534I
a = 2.16216 − 4.82053I
b = −0.004791 − 0.715102I
1.10947 − 1.48655I 5.12329 + 2.41841I
6
II. I
u
2
=
hu
2
+b+1, u
5
−2u
4
+5u
3
−6u
2
+3a+6u−1, u
6
−u
5
+5u
4
−4u
3
+7u
2
−2u+3i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
1
=
u
2
+ 1
u
2
a
10
=
−
1
3
u
5
+
2
3
u
4
+ ··· − 2u +
1
3
−u
2
− 1
a
9
=
−
1
3
u
5
+
2
3
u
4
+ ··· − 2u +
1
3
1
3
u
5
− u
4
+ ··· +
5
3
u − 2
a
8
=
−
1
3
u
4
−
5
3
u
2
−
1
3
u −
5
3
1
3
u
5
− u
4
+ ··· +
5
3
u − 2
a
12
=
−
1
3
u
5
+
2
3
u
4
+ ··· − 2u +
7
3
−
2
3
u
5
+ u
4
+ ··· −
4
3
u + 1
a
4
=
2
3
u
5
+
1
3
u
4
+ ··· +
2
3
u +
5
3
2
3
u
5
+
7
3
u
3
−
1
3
u
2
+
4
3
u − 1
a
7
=
−
2
3
u
5
+
1
3
u
4
+ ··· − u −
1
3
u
3
+ u
2
+ 2u + 1
a
6
=
u
u
3
+ u
a
11
=
−u
5
+
5
3
u
4
+ ··· −
10
3
u +
10
3
−
2
3
u
5
+ u
4
+ ··· −
4
3
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
5
− 3u
4
+ 5u
3
− 11u
2
+ 8u − 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
− 9u
5
+ 31u
4
− 56u
3
+ 63u
2
− 38u + 9
c
2
, c
8
u
6
− u
5
+ 5u
4
− 4u
3
+ 7u
2
− 2u + 3
c
3
, c
5
u
6
+ u
5
+ 5u
4
+ 4u
3
+ 7u
2
+ 2u + 3
c
4
u
6
+ 2u
4
+ 3u
3
+ 2u
2
+ 1
c
6
u
6
− u
5
+ 4u
4
− 2u
3
− 8u
2
+ 6u + 9
c
7
u
6
− 3u
5
+ u
4
− 2u
3
+ 6u
2
+ 5u + 1
c
9
u
6
+ 2u
4
− 3u
3
+ 2u
2
+ 1
c
10
u
6
− 3u
5
+ 5u
3
− u
2
− 2u + 3
c
11
u
6
+ u
5
+ 4u
4
+ 2u
3
− 8u
2
− 6u + 9
c
12
u
6
+ 3u
5
− 5u
3
− u
2
+ 2u + 3
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
− 19y
5
+ 79y
4
+ 104y
3
+ 271y
2
− 310y + 81
c
2
, c
3
, c
5
c
8
y
6
+ 9y
5
+ 31y
4
+ 56y
3
+ 63y
2
+ 38y + 9
c
4
, c
9
y
6
+ 4y
5
+ 8y
4
+ y
3
+ 8y
2
+ 4y + 1
c
6
, c
11
y
6
+ 7y
5
− 4y
4
− 38y
3
+ 160y
2
− 180y + 81
c
7
y
6
− 7y
5
+ y
4
+ 40y
3
+ 58y
2
− 13y + 1
c
10
, c
12
y
6
− 9y
5
+ 28y
4
− 31y
3
+ 21y
2
− 10y + 9
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.615293 + 1.007340I
a = −0.488052 − 0.086507I
b = −0.363854 − 1.239620I
10.45590 + 2.33911I 2.00744 − 2.34673I
u = 0.615293 − 1.007340I
a = −0.488052 + 0.086507I
b = −0.363854 + 1.239620I
10.45590 − 2.33911I 2.00744 + 2.34673I
u = −0.061440 + 0.817267I
a = −0.744380 − 0.966777I
b = −0.335850 + 0.100426I
− 2.22275I 0. + 4.90360I
u = −0.061440 − 0.817267I
a = −0.744380 + 0.966777I
b = −0.335850 − 0.100426I
2.22275I 0. − 4.90360I
u = −0.05385 + 1.78958I
a = 0.899099 + 0.320901I
b = 2.19970 + 0.19275I
−10.45590 − 2.33911I −2.00744 + 2.34673I
u = −0.05385 − 1.78958I
a = 0.899099 − 0.320901I
b = 2.19970 − 0.19275I
−10.45590 + 2.33911I −2.00744 − 2.34673I
10
III. I
u
3
= hb − u, a − u, u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
−u − 1
a
1
=
−u
−u − 1
a
10
=
u
u
a
9
=
u
u + 1
a
8
=
2u + 1
u + 1
a
12
=
1
0
a
4
=
u
u
a
7
=
u
u + 1
a
6
=
u
u + 1
a
11
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
10
u
2
− u + 1
c
2
, c
7
, c
8
c
9
, c
12
u
2
+ u + 1
c
6
, c
11
u
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
7
c
8
, c
9
, c
10
c
12
y
2
+ y + 1
c
6
, c
11
y
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −0.500000 + 0.866025I
b = −0.500000 + 0.866025I
0 0
u = −0.500000 − 0.866025I
a = −0.500000 − 0.866025I
b = −0.500000 − 0.866025I
0 0
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)(u
6
− 9u
5
+ 31u
4
− 56u
3
+ 63u
2
− 38u + 9)
· (u
20
+ 27u
19
+ ··· − 27u + 1)
c
2
(u
2
+ u + 1)(u
6
− u
5
+ 5u
4
− 4u
3
+ 7u
2
− 2u + 3)
· (u
20
+ u
19
+ ··· − 11u + 1)
c
3
(u
2
− u + 1)(u
6
+ u
5
+ 5u
4
+ 4u
3
+ 7u
2
+ 2u + 3)
· (u
20
− u
19
+ ··· + 11u + 1)
c
4
(u
2
− u + 1)(u
6
+ 2u
4
+ ··· + 2u
2
+ 1)(u
20
+ 2u
19
+ ··· − 27u + 51)
c
5
(u
2
− u + 1)(u
6
+ u
5
+ 5u
4
+ 4u
3
+ 7u
2
+ 2u + 3)
· (u
20
+ u
19
+ ··· − 11u + 1)
c
6
u
2
(u
6
− u
5
+ ··· + 6u + 9)(u
20
+ 16u
18
+ ··· − 16u + 52)
c
7
(u
2
+ u + 1)(u
6
− 3u
5
+ u
4
− 2u
3
+ 6u
2
+ 5u + 1)
· (u
20
− 3u
19
+ ··· + 109u
2
+ 21)
c
8
(u
2
+ u + 1)(u
6
− u
5
+ 5u
4
− 4u
3
+ 7u
2
− 2u + 3)
· (u
20
− u
19
+ ··· + 11u + 1)
c
9
(u
2
+ u + 1)(u
6
+ 2u
4
+ ··· + 2u
2
+ 1)(u
20
− 2u
19
+ ··· + 27u + 51)
c
10
(u
2
− u + 1)(u
6
− 3u
5
+ ··· − 2u + 3)(u
20
+ 5u
19
+ ··· + 57u + 7)
c
11
u
2
(u
6
+ u
5
+ ··· − 6u + 9)(u
20
+ 16u
18
+ ··· + 16u + 52)
c
12
(u
2
+ u + 1)(u
6
+ 3u
5
+ ··· + 2u + 3)(u
20
− 5u
19
+ ··· − 57u + 7)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)(y
6
− 19y
5
+ 79y
4
+ 104y
3
+ 271y
2
− 310y + 81)
· (y
20
− 61y
19
+ ··· + 169y + 1)
c
2
, c
3
, c
5
c
8
(y
2
+ y + 1)(y
6
+ 9y
5
+ 31y
4
+ 56y
3
+ 63y
2
+ 38y + 9)
· (y
20
+ 27y
19
+ ··· − 27y + 1)
c
4
, c
9
(y
2
+ y + 1)(y
6
+ 4y
5
+ 8y
4
+ y
3
+ 8y
2
+ 4y + 1)
· (y
20
+ 18y
19
+ ··· + 17835y + 2601)
c
6
, c
11
y
2
(y
6
+ 7y
5
− 4y
4
− 38y
3
+ 160y
2
− 180y + 81)
· (y
20
+ 32y
19
+ ··· + 7024y + 2704)
c
7
(y
2
+ y + 1)(y
6
− 7y
5
+ y
4
+ 40y
3
+ 58y
2
− 13y + 1)
· (y
20
− 33y
19
+ ··· + 4578y + 441)
c
10
, c
12
(y
2
+ y + 1)(y
6
− 9y
5
+ 28y
4
− 31y
3
+ 21y
2
− 10y + 9)
· (y
20
− 15y
19
+ ··· − 1835y + 49)
16
