12n
0221
(K12n
0221
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 11 12 10 3 11 8 1 6
Solving Sequence
2,5 3,11
6 1
4,9
8 10 12 7
c
2
c
5
c
1
c
4
c
8
c
10
c
12
c
6
c
3
, c
7
, c
9
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h2258808925u
16
+ 15860359921u
15
+ ··· + 96479313856d − 115431185160,
154139657205u
16
+ 1193150529938u
15
+ ··· + 2508462160256c − 2116899433348,
4410667u
16
+ 32637095u
15
+ ··· + 83243584b − 126220048,
7888753u
16
+ 58699357u
15
+ ··· + 83243584a − 107418368, u
17
+ 8u
16
+ ··· − 8u − 16i
I
u
2
= hd − a, c − a, b − a, a
2
− a + 1, u − 1i
I
u
3
= hd + 1, c, b − 1, a − 1, u − 1i
I
u
4
= hda − ca + 1, c
2
− c + 1, b − a, u − 1i
I
v
1
= hc, d − a − 1, b, a
2
+ a + 1, v − 1i
* 4 irreducible components of dim
C
= 0, with total 22 representations.
* 1 irreducible components of dim
C
= 1
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
= h2.26 × 10
9
u
16
+ 1.59 × 10
10
u
15
+ · · · + 9.65 × 10
10
d − 1.15 ×
10
11
, 1.54 × 10
11
u
16
+ 1.19 × 10
12
u
15
+ · · · + 2.51 × 10
12
c − 2.12 × 10
12
, 4.41 ×
10
6
u
16
+ 3.26 × 10
7
u
15
+ · · · + 8.32 × 10
7
b − 1.26 × 10
8
, 7.89 × 10
6
u
16
+
5.87 × 10
7
u
15
+ · · · + 8.32 × 10
7
a − 1.07 × 10
8
, u
17
+ 8u
16
+ · · · − 8u − 16i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u
2
a
11
=
−0.0614479u
16
− 0.475650u
15
+ ··· − 2.87951u + 0.843903
−0.0234124u
16
− 0.164391u
15
+ ··· + 0.757508u + 1.19643
a
6
=
0.0269638u
16
+ 0.213054u
15
+ ··· + 1.26938u + 0.117878
−0.00435471u
16
− 0.0521688u
15
+ ··· + 0.124403u − 0.745012
a
1
=
−u
2
+ 1
−u
4
a
4
=
u
u
a
9
=
−0.0947671u
16
− 0.705152u
15
+ ··· + 0.385893u + 1.29041
−0.0529851u
16
− 0.392067u
15
+ ··· − 0.532273u + 1.51627
a
8
=
−0.115939u
16
− 0.867332u
15
+ ··· + 0.946013u + 1.95892
−0.0618930u
16
− 0.460152u
15
+ ··· − 0.813462u + 1.63140
a
10
=
−0.118012u
16
− 0.895172u
15
+ ··· − 0.266541u + 1.78486
−0.0519178u
16
− 0.373867u
15
+ ··· + 0.197985u + 1.94957
a
12
=
−0.0656989u
16
− 0.494197u
15
+ ··· − 2.08468u + 1.58103
−0.0178046u
16
− 0.118986u
15
+ ··· + 0.894693u + 1.10503
a
7
=
0.0678023u
16
+ 0.515689u
15
+ ··· + 1.99680u − 0.449277
0.0201677u
16
+ 0.140474u
15
+ ··· + 0.214188u − 1.03624
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
3288027181
156778885016
u
16
−
220303787499
627115540064
u
15
+ ··· −
1274298730399
156778885016
u −
92162275708
19597360627
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
− 6u
16
+ ··· + 32u + 256
c
2
, c
4
u
17
− 8u
16
+ ··· − 8u + 16
c
3
, c
8
u
17
+ u
16
+ ··· − 1024u + 512
c
5
u
17
+ 14u
16
+ ··· + 6768u + 2592
c
6
, c
12
u
17
− 5u
16
+ ··· − 11u
2
+ 4
c
7
, c
10
u
17
+ 8u
16
+ ··· − 8u + 16
c
9
u
17
− 34u
16
+ ··· + 6176u − 256
c
11
u
17
− 15u
16
+ ··· + 88u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
+ 66y
16
+ ··· + 2613760y − 65536
c
2
, c
4
y
17
+ 6y
16
+ ··· + 32y − 256
c
3
, c
8
y
17
+ 81y
16
+ ··· − 524288y − 262144
c
5
y
17
− 66y
16
+ ··· + 36764928y − 6718464
c
6
, c
12
y
17
+ 15y
16
+ ··· + 88y − 16
c
7
, c
10
y
17
− 34y
16
+ ··· + 6176y − 256
c
9
y
17
− 94y
16
+ ··· + 7397888y − 65536
c
11
y
17
− 21y
16
+ ··· + 36640y − 256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.789321
a = 0.386224
b = −0.304855
c = 0.374974
d = −0.586930
−1.13318 −9.61860
u = 1.281020 + 0.078323I
a = 0.027802 − 0.314358I
b = −0.060236 + 0.400521I
c = 0.519378 − 0.243814I
d = 0.992769 + 0.281518I
−0.72956 − 1.37071I −0.698150 + 0.213889I
u = 1.281020 − 0.078323I
a = 0.027802 + 0.314358I
b = −0.060236 − 0.400521I
c = 0.519378 + 0.243814I
d = 0.992769 − 0.281518I
−0.72956 + 1.37071I −0.698150 − 0.213889I
u = −0.709544 + 0.075286I
a = 0.27087 + 2.81412I
b = 0.40406 + 1.97635I
c = 0.41875 + 1.62653I
d = 0.386110 + 1.184900I
−0.79868 + 2.33972I 0.33078 − 5.26516I
u = −0.709544 − 0.075286I
a = 0.27087 − 2.81412I
b = 0.40406 − 1.97635I
c = 0.41875 − 1.62653I
d = 0.386110 − 1.184900I
−0.79868 − 2.33972I 0.33078 + 5.26516I
u = 0.491842 + 0.197993I
a = −0.746901 + 0.354453I
b = 0.437536 − 0.026454I
c = −0.68906 + 1.41559I
d = 1.212590 − 0.601554I
0.77904 − 2.74622I −2.48507 + 7.16740I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.491842 − 0.197993I
a = −0.746901 − 0.354453I
b = 0.437536 + 0.026454I
c = −0.68906 − 1.41559I
d = 1.212590 + 0.601554I
0.77904 + 2.74622I −2.48507 − 7.16740I
u = −0.118015 + 0.350813I
a = −0.50791 + 1.75422I
b = 0.555461 + 0.385207I
c = 1.154330 + 0.052946I
d = 0.083184 + 0.147223I
1.75773 + 0.71028I 3.71531 + 0.02644I
u = −0.118015 − 0.350813I
a = −0.50791 − 1.75422I
b = 0.555461 − 0.385207I
c = 1.154330 − 0.052946I
d = 0.083184 − 0.147223I
1.75773 − 0.71028I 3.71531 − 0.02644I
u = −1.65818 + 0.90820I
a = 0.976512 + 0.609189I
b = 2.17250 + 0.12328I
c = −0.94081 − 1.33349I
d = −5.94499 − 2.01193I
18.4182 + 12.9335I −1.01650 − 5.27491I
u = −1.65818 − 0.90820I
a = 0.976512 − 0.609189I
b = 2.17250 − 0.12328I
c = −0.94081 + 1.33349I
d = −5.94499 + 2.01193I
18.4182 − 12.9335I −1.01650 + 5.27491I
u = −1.62328 + 1.28695I
a = −0.780321 − 0.614432I
b = −2.05742 + 0.00684I
c = 0.67125 + 1.78783I
d = 8.16028 + 1.30179I
15.3110 + 5.6503I −2.10303 − 1.68119I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.62328 − 1.28695I
a = −0.780321 + 0.614432I
b = −2.05742 − 0.00684I
c = 0.67125 − 1.78783I
d = 8.16028 − 1.30179I
15.3110 − 5.6503I −2.10303 + 1.68119I
u = −0.77580 + 2.21598I
a = −0.394911 − 0.622157I
b = −1.68506 + 0.39245I
c = −1.69156 − 0.56408I
d = 4.42924 + 7.49395I
9.63429 + 3.26152I 0.10201 − 1.44169I
u = −0.77580 − 2.21598I
a = −0.394911 + 0.622157I
b = −1.68506 − 0.39245I
c = −1.69156 + 0.56408I
d = 4.42924 − 7.49395I
9.63429 − 3.26152I 0.10201 + 1.44169I
u = −1.28271 + 2.40373I
a = 0.461749 + 0.538038I
b = 1.88559 − 0.41977I
c = −0.25477 − 2.11813I
d = −12.0257 + 7.8857I
−17.4865 − 1.7702I − 60.10 + 0.657690I
u = −1.28271 − 2.40373I
a = 0.461749 − 0.538038I
b = 1.88559 + 0.41977I
c = −0.25477 + 2.11813I
d = −12.0257 − 7.8857I
−17.4865 + 1.7702I − 60.10 − 0.657690I
7
II. I
u
2
= hd − a, c − a, b − a, a
2
− a + 1, u − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
3
=
1
1
a
11
=
a
a
a
6
=
a − 1
a
a
1
=
0
−1
a
4
=
1
1
a
9
=
a
a
a
8
=
a
a
a
10
=
a
a
a
12
=
a
0
a
7
=
a
a
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4a − 7
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
7
, c
8
c
9
, c
10
u
2
c
4
(u + 1)
2
c
5
, c
11
, c
12
u
2
+ u + 1
c
6
u
2
− u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
2
c
3
, c
7
, c
8
c
9
, c
10
y
2
c
5
, c
6
, c
11
c
12
y
2
+ y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.500000 + 0.866025I
b = 0.500000 + 0.866025I
c = 0.500000 + 0.866025I
d = 0.500000 + 0.866025I
−1.64493 + 2.02988I −9.00000 − 3.46410I
u = 1.00000
a = 0.500000 − 0.866025I
b = 0.500000 − 0.866025I
c = 0.500000 − 0.866025I
d = 0.500000 − 0.866025I
−1.64493 − 2.02988I −9.00000 + 3.46410I
11
III. I
u
3
= hd + 1, c, b − 1, a − 1, u − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
3
=
1
1
a
11
=
0
−1
a
6
=
0
1
a
1
=
0
−1
a
4
=
1
1
a
9
=
1
1
a
8
=
1
1
a
10
=
1
0
a
12
=
0
−1
a
7
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
10
u − 1
c
3
, c
5
, c
6
c
8
, c
11
, c
12
u
c
4
, c
7
, c
9
u + 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
7
, c
9
, c
10
y − 1
c
3
, c
5
, c
6
c
8
, c
11
, c
12
y
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 1.00000
c = 0
d = −1.00000
0 0
15
IV. I
u
4
= hda − ca + 1, c
2
− c + 1, b − a, u − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
3
=
1
1
a
11
=
c
d
a
6
=
c − 1
dc + 1
a
1
=
0
−1
a
4
=
1
1
a
9
=
a
a
a
8
=
a
a
a
10
=
c + a
d + a
a
12
=
c
d − c
a
7
=
c
d
(ii) Obstruction class = −1
(iii) Cusp Shapes = d
2
− 2dc + a
2
− 3c − 1
(iv) u-Polynomials at the component : It cannot be defined for a positive
dimension component.
(v) Riley Polynomials at the component : It cannot be defined for a positive
dimension component.
16
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
c = ···
d = ···
− 2.02988I −0.06692 − 3.42770I
17
V. I
v
1
= hc, d − a − 1, b, a
2
+ a + 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
1
0
a
3
=
1
0
a
11
=
0
a + 1
a
6
=
1
−a
a
1
=
1
0
a
4
=
1
0
a
9
=
a
0
a
8
=
a
0
a
10
=
a
a + 1
a
12
=
a + 1
a + 1
a
7
=
0
−a − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4a + 1
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
8
u
2
c
5
, c
12
u
2
− u + 1
c
6
, c
11
u
2
+ u + 1
c
7
, c
9
(u + 1)
2
c
10
(u − 1)
2
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
8
y
2
c
5
, c
6
, c
11
c
12
y
2
+ y + 1
c
7
, c
9
, c
10
(y − 1)
2
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = −0.500000 + 0.866025I
b = 0
c = 0
d = 0.500000 + 0.866025I
1.64493 + 2.02988I 3.00000 − 3.46410I
v = 1.00000
a = −0.500000 − 0.866025I
b = 0
c = 0
d = 0.500000 − 0.866025I
1.64493 − 2.02988I 3.00000 + 3.46410I
21
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
2
(u − 1)
3
(u
17
− 6u
16
+ ··· + 32u + 256)
c
2
u
2
(u − 1)
3
(u
17
− 8u
16
+ ··· − 8u + 16)
c
3
, c
8
u
5
(u
17
+ u
16
+ ··· − 1024u + 512)
c
4
u
2
(u + 1)
3
(u
17
− 8u
16
+ ··· − 8u + 16)
c
5
u(u
2
− u + 1)(u
2
+ u + 1)(u
17
+ 14u
16
+ ··· + 6768u + 2592)
c
6
, c
12
u(u
2
− u + 1)(u
2
+ u + 1)(u
17
− 5u
16
+ ··· − 11u
2
+ 4)
c
7
u
2
(u + 1)
3
(u
17
+ 8u
16
+ ··· − 8u + 16)
c
9
u
2
(u + 1)
3
(u
17
− 34u
16
+ ··· + 6176u − 256)
c
10
u
2
(u − 1)
3
(u
17
+ 8u
16
+ ··· − 8u + 16)
c
11
u(u
2
+ u + 1)
2
(u
17
− 15u
16
+ ··· + 88u + 16)
22
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
2
(y − 1)
3
(y
17
+ 66y
16
+ ··· + 2613760y − 65536)
c
2
, c
4
y
2
(y − 1)
3
(y
17
+ 6y
16
+ ··· + 32y − 256)
c
3
, c
8
y
5
(y
17
+ 81y
16
+ ··· − 524288y − 262144)
c
5
y(y
2
+ y + 1)
2
(y
17
− 66y
16
+ ··· + 3.67649 × 10
7
y − 6718464)
c
6
, c
12
y(y
2
+ y + 1)
2
(y
17
+ 15y
16
+ ··· + 88y − 16)
c
7
, c
10
y
2
(y − 1)
3
(y
17
− 34y
16
+ ··· + 6176y − 256)
c
9
y
2
(y − 1)
3
(y
17
− 94y
16
+ ··· + 7397888y − 65536)
c
11
y(y
2
+ y + 1)
2
(y
17
− 21y
16
+ ··· + 36640y − 256)
23
