12n
0726
(K12n
0726
)
A knot diagram
1
Linearized knot diagam
4 6 7 9 3 2 11 12 4 1 9 8
Solving Sequence
8,12 4,9
5 1 2 11 7 3 6 10
c
8
c
4
c
12
c
1
c
11
c
7
c
3
c
6
c
10
c
2
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
5
+ u
4
+ 2u
3
+ 2u
2
+ b, −u
5
− u
4
− 3u
3
− 2u
2
+ a − 2u,
u
10
+ u
9
+ 6u
8
+ 5u
7
+ 12u
6
+ 8u
5
+ 8u
4
+ 3u
3
+ u
2
− 2u + 1i
I
u
2
= h−2u
29
− 3u
28
+ ··· + 2b + 10, −13u
29
− 37u
28
+ ··· + 2a + 42, u
30
+ 3u
29
+ ··· − 8u − 1i
I
u
3
= hu
2
+ b, a + 1, u
3
− u
2
+ 2u − 1i
I
u
4
= h−u
2
a + b, −u
2
a + a
2
+ u
2
− 2a + 2, u
3
− u
2
+ 2u − 1i
* 4 irreducible components of dim
C
= 0, with total 49 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
= hu
5
+u
4
+2u
3
+2u
2
+b, −u
5
−u
4
−3u
3
−2u
2
+a−2u, u
10
+u
9
+· · ·−2u+1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
u
5
+ u
4
+ 3u
3
+ 2u
2
+ 2u
−u
5
− u
4
− 2u
3
− 2u
2
a
9
=
1
u
2
a
5
=
−u
7
− u
6
− 3u
5
− 2u
4
− u
3
+ 2u
−u
9
− u
8
− 4u
7
− 3u
6
− 5u
5
− 3u
4
− 2u
3
− 2u
2
a
1
=
−u
u
a
2
=
−u
9
− u
8
− 5u
7
− 4u
6
− 8u
5
− 4u
4
− 4u
3
− u
u
9
+ u
8
+ 4u
7
+ 4u
6
+ 4u
5
+ 4u
4
+ u
a
11
=
u
u
3
+ u
a
7
=
−u
4
− u
2
+ 1
−u
6
− 2u
4
− u
2
a
3
=
u
7
+ 3u
5
+ u
4
+ 3u
3
+ 2u
2
+ u
u
9
+ 3u
7
+ 2u
5
− u
4
− u
3
− 2u
2
a
6
=
−u
8
− u
7
− 4u
6
− 4u
5
− 5u
4
− 3u
3
− u
2
+ 2u
2u
8
+ u
7
+ 7u
6
+ 4u
5
+ 6u
4
+ 3u
3
− u
2
− 2u + 1
a
10
=
u
5
+ 2u
3
+ u
−u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
9
+ 4u
8
+ 22u
7
+ 22u
6
+ 42u
5
+ 40u
4
+ 30u
3
+ 20u
2
+ 8u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
10
− u
9
+ 8u
8
− 7u
7
+ 18u
6
− 18u
5
+ 8u
4
− 13u
3
+ 7u
2
+ 1
c
2
, c
5
, c
6
c
8
, c
11
, c
12
u
10
− u
9
+ 6u
8
− 5u
7
+ 12u
6
− 8u
5
+ 8u
4
− 3u
3
+ u
2
+ 2u + 1
c
3
, c
7
u
10
+ u
9
+ 4u
8
+ 4u
7
+ 21u
6
− 9u
5
+ 33u
4
+ 5u
3
+ 7u
2
+ 3u + 2
c
4
, c
9
u
10
+ 7u
9
+ ··· + 24u + 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
10
+ 15y
9
+ ··· + 14y + 1
c
2
, c
5
, c
6
c
8
, c
11
, c
12
y
10
+ 11y
9
+ ··· − 2y + 1
c
3
, c
7
y
10
+ 7y
9
+ ··· + 19y + 4
c
4
, c
9
y
10
− 7y
9
+ ··· + 256y + 64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.728898 + 0.479191I
a = −0.415541 + 1.217980I
b = −0.927397 + 0.394143I
5.65760 + 4.71262I −6.31236 − 5.61759I
u = −0.728898 − 0.479191I
a = −0.415541 − 1.217980I
b = −0.927397 − 0.394143I
5.65760 − 4.71262I −6.31236 + 5.61759I
u = 0.066306 + 1.207890I
a = −0.854695 − 0.475151I
b = 0.697376 + 1.144550I
5.29548 − 2.05211I −3.55200 + 3.27198I
u = 0.066306 − 1.207890I
a = −0.854695 + 0.475151I
b = 0.697376 − 1.144550I
5.29548 + 2.05211I −3.55200 − 3.27198I
u = 0.11337 + 1.49042I
a = 1.075600 − 0.665737I
b = −1.60291 + 0.39329I
11.32540 − 4.10290I 0.47358 + 2.87242I
u = 0.11337 − 1.49042I
a = 1.075600 + 0.665737I
b = −1.60291 − 0.39329I
11.32540 + 4.10290I 0.47358 − 2.87242I
u = −0.28831 + 1.50977I
a = −2.00632 + 0.94362I
b = 3.37727 − 0.98896I
18.5056 + 12.2668I −0.40879 − 5.71170I
u = −0.28831 − 1.50977I
a = −2.00632 − 0.94362I
b = 3.37727 + 0.98896I
18.5056 − 12.2668I −0.40879 + 5.71170I
u = 0.337535 + 0.237080I
a = 0.700954 + 1.016800I
b = −0.044343 − 0.474934I
−0.483217 − 0.888721I −8.20043 + 7.80792I
u = 0.337535 − 0.237080I
a = 0.700954 − 1.016800I
b = −0.044343 + 0.474934I
−0.483217 + 0.888721I −8.20043 − 7.80792I
5
II. I
u
2
= h−2u
29
− 3u
28
+ · · · + 2b + 10, −13u
29
− 37u
28
+ · · · + 2a +
42, u
30
+ 3u
29
+ · · · − 8u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
13
2
u
29
+
37
2
u
28
+ ··· −
209
2
u − 21
u
29
+
3
2
u
28
+ ··· −
43
2
u − 5
a
9
=
1
u
2
a
5
=
15
2
u
29
+
43
2
u
28
+ ··· −
249
2
u − 25
u
29
+
3
2
u
28
+ ··· −
41
2
u − 5
a
1
=
−u
u
a
2
=
−u
29
−
5
2
u
28
+ ··· +
11
2
u +
5
2
−
1
2
u
27
− u
26
+ ··· + 3u +
1
2
a
11
=
u
u
3
+ u
a
7
=
−u
4
− u
2
+ 1
−u
6
− 2u
4
− u
2
a
3
=
8u
29
+ 23u
28
+ ··· − 142u −
57
2
5
2
u
29
+ 5u
28
+ ··· − 20u −
9
2
a
6
=
7u
29
+ 19u
28
+ ··· − 85u − 13
1
2
u
29
−
1
2
u
28
+ ··· −
31
2
u − 3
a
10
=
u
5
+ 2u
3
+ u
−u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
11
2
u
29
+ 15u
28
+ ··· − 99u −
51
2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
30
− 3u
29
+ ··· − 18u
2
+ 1
c
2
, c
5
, c
6
c
8
, c
11
, c
12
u
30
− 3u
29
+ ··· + 8u − 1
c
3
, c
7
u
30
+ 3u
29
+ ··· + 1074u − 153
c
4
, c
9
(u
15
− 3u
14
+ ··· − 12u + 8)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
30
+ 37y
29
+ ··· − 36y + 1
c
2
, c
5
, c
6
c
8
, c
11
, c
12
y
30
+ 29y
29
+ ··· − 20y + 1
c
3
, c
7
y
30
+ 17y
29
+ ··· − 192024y + 23409
c
4
, c
9
(y
15
− 21y
14
+ ··· + 784y −64)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.705981 + 0.612665I
a = −0.189821 + 0.894198I
b = −1.092990 + 0.473497I
12.57910 − 3.33907I −2.38574 + 0.22991I
u = −0.705981 − 0.612665I
a = −0.189821 − 0.894198I
b = −1.092990 − 0.473497I
12.57910 + 3.33907I −2.38574 − 0.22991I
u = −0.786126 + 0.466474I
a = 0.307960 − 1.352620I
b = 0.918649 − 0.316331I
12.1008 + 8.3364I −3.32084 − 5.47194I
u = −0.786126 − 0.466474I
a = 0.307960 + 1.352620I
b = 0.918649 + 0.316331I
12.1008 − 8.3364I −3.32084 + 5.47194I
u = −0.685541 + 0.538898I
a = 0.386781 − 0.997148I
b = 0.987952 − 0.464825I
5.87309 −5.59057 + 0.I
u = −0.685541 − 0.538898I
a = 0.386781 + 0.997148I
b = 0.987952 + 0.464825I
5.87309 −5.59057 + 0.I
u = 0.713656 + 0.166582I
a = −0.855543 − 0.376613I
b = 0.448037 + 0.280112I
2.81581 − 0.87895I −5.63582 + 0.83931I
u = 0.713656 − 0.166582I
a = −0.855543 + 0.376613I
b = 0.448037 − 0.280112I
2.81581 + 0.87895I −5.63582 − 0.83931I
u = 0.243602 + 1.279880I
a = 0.169871 + 0.426938I
b = 0.032440 − 0.551805I
2.51678 − 3.17894I 0.37815 + 5.88971I
u = 0.243602 − 1.279880I
a = 0.169871 − 0.426938I
b = 0.032440 + 0.551805I
2.51678 + 3.17894I 0.37815 − 5.88971I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.302233 + 0.572979I
a = −0.891160 − 0.823170I
b = 0.029484 + 0.794061I
4.68149 − 2.48936I −2.42897 + 4.40087I
u = 0.302233 − 0.572979I
a = −0.891160 + 0.823170I
b = 0.029484 − 0.794061I
4.68149 + 2.48936I −2.42897 − 4.40087I
u = −0.029336 + 1.362430I
a = 1.53009 + 0.04314I
b = −2.06494 − 0.86724I
2.81581 + 0.87895I −5.63582 − 0.83931I
u = −0.029336 − 1.362430I
a = 1.53009 − 0.04314I
b = −2.06494 + 0.86724I
2.81581 − 0.87895I −5.63582 + 0.83931I
u = 0.635625
a = 0.526188
b = −0.268208
−1.48208 −4.72100
u = 0.315927 + 1.335280I
a = 0.056628 − 0.761520I
b = −0.400717 + 0.793524I
7.52431 − 4.63680I 0. + 2.51110I
u = 0.315927 − 1.335280I
a = 0.056628 + 0.761520I
b = −0.400717 − 0.793524I
7.52431 + 4.63680I 0. − 2.51110I
u = 0.099338 + 1.386390I
a = −0.972274 + 0.248617I
b = 1.323920 + 0.164440I
4.68149 − 2.48936I −2.42897 + 4.40087I
u = 0.099338 − 1.386390I
a = −0.972274 − 0.248617I
b = 1.323920 − 0.164440I
4.68149 + 2.48936I −2.42897 − 4.40087I
u = −0.084417 + 1.399330I
a = −1.87882 + 0.15021I
b = 2.75314 + 0.68684I
7.52431 + 4.63680I 0. − 2.51110I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.084417 − 1.399330I
a = −1.87882 − 0.15021I
b = 2.75314 − 0.68684I
7.52431 − 4.63680I 0. + 2.51110I
u = −0.26139 + 1.50518I
a = 2.03229 − 0.92308I
b = −3.40445 + 0.88020I
12.1008 + 8.3364I 0. − 5.47194I
u = −0.26139 − 1.50518I
a = 2.03229 + 0.92308I
b = −3.40445 − 0.88020I
12.1008 − 8.3364I 0. + 5.47194I
u = −0.23109 + 1.51730I
a = −2.02333 + 0.87807I
b = 3.32265 − 0.75494I
12.57910 + 3.33907I 0
u = −0.23109 − 1.51730I
a = −2.02333 − 0.87807I
b = 3.32265 + 0.75494I
12.57910 − 3.33907I 0
u = −0.21368 + 1.55326I
a = 1.97146 − 0.87493I
b = −3.15532 + 0.76228I
19.7380 0
u = −0.21368 − 1.55326I
a = 1.97146 + 0.87493I
b = −3.15532 − 0.76228I
19.7380 0
u = −0.354580 + 0.145881I
a = 2.47267 − 0.38301I
b = 0.785485 − 0.312110I
2.51678 + 3.17894I 0.37815 − 5.88971I
u = −0.354580 − 0.145881I
a = 2.47267 + 0.38301I
b = 0.785485 + 0.312110I
2.51678 − 3.17894I 0.37815 + 5.88971I
u = −0.280853
a = −2.75980
b = −0.698487
−1.48208 −4.72100
11
III. I
u
3
= hu
2
+ b, a + 1, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
−1
−u
2
a
9
=
1
u
2
a
5
=
−1
−u
2
a
1
=
−u
u
a
2
=
−u
2
− 1
u
2
a
11
=
u
u
2
− u + 1
a
7
=
u
−u
a
3
=
u − 2
−u
2
− u + 1
a
6
=
−u
2
+ 2u − 1
u
2
− 3u + 1
a
10
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u
2
+ 8u − 20
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
10
u
3
+ u
2
− 1
c
2
, c
8
u
3
− u
2
+ 2u − 1
c
4
, c
9
u
3
c
5
, c
6
, c
11
c
12
u
3
+ u
2
+ 2u + 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
7
c
10
y
3
− y
2
+ 2y −1
c
2
, c
5
, c
6
c
8
, c
11
, c
12
y
3
+ 3y
2
+ 2y −1
c
4
, c
9
y
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −1.00000
b = 1.66236 − 0.56228I
6.04826 − 5.65624I −4.98049 + 5.95889I
u = 0.215080 − 1.307140I
a = −1.00000
b = 1.66236 + 0.56228I
6.04826 + 5.65624I −4.98049 − 5.95889I
u = 0.569840
a = −1.00000
b = −0.324718
−2.22691 −18.0390
15
IV. I
u
4
= h−u
2
a + b, −u
2
a + a
2
+ u
2
− 2a + 2, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
a
u
2
a
a
9
=
1
u
2
a
5
=
a
u
2
a
a
1
=
−u
u
a
2
=
−u
2
a + au + u
2
− 2a − 2u + 2
−au + 2u
a
11
=
u
u
2
− u + 1
a
7
=
u
−u
a
3
=
−au + 2a
u
2
a + au − a
a
6
=
−2u
2
a + 2au + 3u
2
− 2a − u + 4
2u
2
a − 2au − u
2
+ a
a
10
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
2
a − 5au − 3u
2
+ 3a + 3u − 12
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
10
(u
3
+ u
2
− 1)
2
c
2
, c
8
(u
3
− u
2
+ 2u − 1)
2
c
4
, c
9
u
6
c
5
, c
6
, c
11
c
12
(u
3
+ u
2
+ 2u + 1)
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
7
c
10
(y
3
− y
2
+ 2y −1)
2
c
2
, c
5
, c
6
c
8
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
c
4
, c
9
y
6
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −0.162359 + 0.986732I
b = −0.28492 − 1.73159I
6.04826 −6 − 1.085931 + 0.10I
u = 0.215080 + 1.307140I
a = 0.500000 − 0.424452I
b = −0.592519 + 0.986732I
1.91067 − 2.82812I −9.95703 + 1.11003I
u = 0.215080 − 1.307140I
a = −0.162359 − 0.986732I
b = −0.28492 + 1.73159I
6.04826 −6 − 1.085931 + 0.10I
u = 0.215080 − 1.307140I
a = 0.500000 + 0.424452I
b = −0.592519 − 0.986732I
1.91067 + 2.82812I −9.95703 − 1.11003I
u = 0.569840
a = 1.16236 + 0.98673I
b = 0.377439 + 0.320410I
1.91067 − 2.82812I −9.95703 + 1.11003I
u = 0.569840
a = 1.16236 − 0.98673I
b = 0.377439 − 0.320410I
1.91067 + 2.82812I −9.95703 − 1.11003I
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
10
(u
3
+ u
2
− 1)
3
· (u
10
− u
9
+ 8u
8
− 7u
7
+ 18u
6
− 18u
5
+ 8u
4
− 13u
3
+ 7u
2
+ 1)
· (u
30
− 3u
29
+ ··· − 18u
2
+ 1)
c
2
, c
8
(u
3
− u
2
+ 2u − 1)
3
· (u
10
− u
9
+ 6u
8
− 5u
7
+ 12u
6
− 8u
5
+ 8u
4
− 3u
3
+ u
2
+ 2u + 1)
· (u
30
− 3u
29
+ ··· + 8u − 1)
c
3
, c
7
(u
3
+ u
2
− 1)
3
· (u
10
+ u
9
+ 4u
8
+ 4u
7
+ 21u
6
− 9u
5
+ 33u
4
+ 5u
3
+ 7u
2
+ 3u + 2)
· (u
30
+ 3u
29
+ ··· + 1074u − 153)
c
4
, c
9
u
9
(u
10
+ 7u
9
+ ··· + 24u + 8)(u
15
− 3u
14
+ ··· − 12u + 8)
2
c
5
, c
6
, c
11
c
12
(u
3
+ u
2
+ 2u + 1)
3
· (u
10
− u
9
+ 6u
8
− 5u
7
+ 12u
6
− 8u
5
+ 8u
4
− 3u
3
+ u
2
+ 2u + 1)
· (u
30
− 3u
29
+ ··· + 8u − 1)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
10
((y
3
− y
2
+ 2y −1)
3
)(y
10
+ 15y
9
+ ··· + 14y + 1)
· (y
30
+ 37y
29
+ ··· − 36y + 1)
c
2
, c
5
, c
6
c
8
, c
11
, c
12
((y
3
+ 3y
2
+ 2y −1)
3
)(y
10
+ 11y
9
+ ··· − 2y + 1)
· (y
30
+ 29y
29
+ ··· − 20y + 1)
c
3
, c
7
((y
3
− y
2
+ 2y −1)
3
)(y
10
+ 7y
9
+ ··· + 19y + 4)
· (y
30
+ 17y
29
+ ··· − 192024y + 23409)
c
4
, c
9
y
9
(y
10
− 7y
9
+ ··· + 256y + 64)(y
15
− 21y
14
+ ··· + 784y −64)
2
21
