12n
0057
(K12n
0057
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 12 10 5 7 9 1 7
Solving Sequence
2,5
3
6,7,10
8 1 4 9 12 11
c
2
c
5
c
7
c
1
c
4
c
9
c
12
c
11
c
3
, c
6
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h769u
16
− 3605u
15
+ ··· + 4864d − 844, 161u
16
− 983u
15
+ ··· + 4864c + 3868,
191u
16
− 846u
15
+ ··· + 4864b + 776, 161u
16
− 983u
15
+ ··· + 4864a + 3868,
u
17
− 5u
16
+ ··· − 11u
2
+ 4i
I
u
2
= hd − u − 1, c, b − u − 1, a, u
2
+ u + 1i
I
u
3
= hd + 2u + 1, c + u, b − u, a − u, u
2
+ u + 1i
I
u
4
= hda + a
2
+ au + a − 1, c + a, b − a − u − 1, u
2
+ u + 1i
I
v
1
= ha, d − 1, c + a − 1, b + 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
= h769u
16
− 3605u
15
+ · · · + 4864d − 844, 161u
16
− 983u
15
+ · · · +
4864c + 3868, 191u
16
− 846u
15
+ · · · + 4864b + 776, 161u
16
− 983u
15
+ · · · +
4864a + 3868, u
17
− 5u
16
+ · · · − 11u
2
+ 4i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
−u
2
a
6
=
u
u
a
7
=
−0.0331003u
16
+ 0.202097u
15
+ ··· + 4.36842u − 0.795230
−0.0392681u
16
+ 0.173931u
15
+ ··· + 1.37253u − 0.159539
a
10
=
−0.0331003u
16
+ 0.202097u
15
+ ··· + 4.36842u − 0.795230
−0.158100u
16
+ 0.741160u
15
+ ··· − 0.475329u + 0.173520
a
8
=
0.0246711u
16
− 0.0435855u
15
+ ··· + 4.48355u − 0.167763
−0.125822u
16
+ 0.625411u
15
+ ··· − 0.166118u + 0.268092
a
1
=
u
2
+ 1
−u
4
a
4
=
u
4
+ u
2
+ 1
u
4
a
9
=
0.0246711u
16
− 0.0435855u
15
+ ··· + 4.48355u − 0.167763
−0.164474u
16
+ 0.808799u
15
+ ··· − 0.0674342u + 0.587171
a
12
=
0.0764803u
16
− 0.397615u
15
+ ··· − 2.61349u + 1.50493
0.0542763u
16
− 0.265419u
15
+ ··· − 0.489309u + 0.0715461
a
11
=
0.0618832u
16
− 0.294613u
15
+ ··· − 1.74178u + 1.47451
0.0807977u
16
− 0.432977u
15
+ ··· − 0.311678u + 0.136513
(ii) Obstruction class = −1
(iii) Cusp Shapes =
409
1216
u
16
−
1133
608
u
15
+ ··· −
4283
304
u −
37
152
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
+ 15u
16
+ ··· + 88u − 16
c
2
, c
5
u
17
+ 5u
16
+ ··· + 11u
2
− 4
c
3
u
17
− 14u
16
+ ··· + 6768u − 2592
c
4
, c
8
u
17
− u
16
+ ··· − 1024u − 512
c
6
, c
12
u
17
+ 8u
16
+ ··· − 8u − 16
c
7
, c
9
u
17
− 8u
16
+ ··· − 8u − 16
c
10
u
17
+ 34u
16
+ ··· + 6176u + 256
c
11
u
17
+ 6u
16
+ ··· + 32u − 256
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
− 21y
16
+ ··· + 36640y − 256
c
2
, c
5
y
17
+ 15y
16
+ ··· + 88y − 16
c
3
y
17
− 66y
16
+ ··· + 36764928y − 6718464
c
4
, c
8
y
17
+ 81y
16
+ ··· − 524288y − 262144
c
6
, c
12
y
17
+ 6y
16
+ ··· + 32y − 256
c
7
, c
9
y
17
− 34y
16
+ ··· + 6176y − 256
c
10
y
17
− 94y
16
+ ··· + 7397888y − 65536
c
11
y
17
+ 66y
16
+ ··· + 2613760y − 65536
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.589168 + 0.828507I
a = −0.334446 + 0.242523I
b = 0.034957 + 0.580654I
c = −0.334446 + 0.242523I
d = −0.698979 − 0.215129I
0.79868 − 2.33972I −0.33078 + 5.26516I
u = −0.589168 − 0.828507I
a = −0.334446 − 0.242523I
b = 0.034957 − 0.580654I
c = −0.334446 − 0.242523I
d = −0.698979 + 0.215129I
0.79868 + 2.33972I −0.33078 − 5.26516I
u = −0.403846 + 0.948035I
a = −0.05430 + 1.74034I
b = 0.19668 + 1.84724I
c = −0.05430 + 1.74034I
d = 0.10664 + 2.23981I
−0.77904 − 2.74622I 2.48507 + 7.16740I
u = −0.403846 − 0.948035I
a = −0.05430 − 1.74034I
b = 0.19668 − 1.84724I
c = −0.05430 − 1.74034I
d = 0.10664 − 2.23981I
−0.77904 + 2.74622I 2.48507 − 7.16740I
u = 0.329450 + 1.030540I
a = −0.533679 − 0.078695I
b = −1.17048 + 1.30416I
c = −0.533679 − 0.078695I
d = −2.21402 + 0.44981I
0.72956 + 1.37071I 0.698150 − 0.213889I
u = 0.329450 − 1.030540I
a = −0.533679 + 0.078695I
b = −1.17048 − 1.30416I
c = −0.533679 + 0.078695I
d = −2.21402 − 0.44981I
0.72956 − 1.37071I 0.698150 + 0.213889I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.349370 + 0.320500I
a = −0.41610 − 1.64751I
b = −0.40657 + 1.44659I
c = −0.41610 − 1.64751I
d = 0.216006 + 0.318534I
−15.3110 − 5.6503I 2.10303 + 1.68119I
u = 1.349370 − 0.320500I
a = −0.41610 + 1.64751I
b = −0.40657 − 1.44659I
c = −0.41610 + 1.64751I
d = 0.216006 − 0.318534I
−15.3110 + 5.6503I 2.10303 − 1.68119I
u = 0.76686 + 1.31677I
a = −1.25346 − 0.81948I
b = −3.59718 − 0.72270I
c = −1.25346 − 0.81948I
d = −2.85722 − 1.77893I
−18.4182 + 12.9335I 1.01650 − 5.27491I
u = 0.76686 − 1.31677I
a = −1.25346 + 0.81948I
b = −3.59718 + 0.72270I
c = −1.25346 + 0.81948I
d = −2.85722 + 1.77893I
−18.4182 − 12.9335I 1.01650 + 5.27491I
u = 0.249371 + 0.383586I
a = 0.211561 + 0.671412I
b = 0.189849 + 0.372765I
c = 0.211561 + 0.671412I
d = 0.330805 − 1.044730I
−1.75773 + 0.71028I −3.71531 + 0.02644I
u = 0.249371 − 0.383586I
a = 0.211561 − 0.671412I
b = 0.189849 − 0.372765I
c = 0.211561 − 0.671412I
d = 0.330805 + 1.044730I
−1.75773 − 0.71028I −3.71531 − 0.02644I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.275145
a = −2.98224
b = −0.717882
c = −2.98224
d = −0.605476
1.13318 9.61860
u = 0.30683 + 1.77436I
a = −0.324822 − 0.574504I
b = −1.79861 − 3.25766I
c = −0.324822 − 0.574504I
d = −0.852724 + 0.332528I
−9.63429 + 3.26152I −0.10201 − 1.44169I
u = 0.30683 − 1.77436I
a = −0.324822 + 0.574504I
b = −1.79861 + 3.25766I
c = −0.324822 + 0.574504I
d = −0.852724 − 0.332528I
−9.63429 − 3.26152I −0.10201 + 1.44169I
u = 0.62871 + 1.82695I
a = 1.196370 + 0.402403I
b = 4.91030 − 0.58153I
c = 1.196370 + 0.402403I
d = 2.77223 + 0.28963I
17.4865 + 1.7702I 0.036073 − 0.657690I
u = 0.62871 − 1.82695I
a = 1.196370 − 0.402403I
b = 4.91030 + 0.58153I
c = 1.196370 − 0.402403I
d = 2.77223 − 0.28963I
17.4865 − 1.7702I 0.036073 + 0.657690I
7
II. I
u
2
= hd − u − 1, c, b − u − 1, a, u
2
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u + 1
a
6
=
u
u
a
7
=
0
u + 1
a
10
=
0
u + 1
a
8
=
0
u + 1
a
1
=
−u
−u
a
4
=
0
u
a
9
=
0
u + 1
a
12
=
−u
1
a
11
=
0
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u + 11
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
u
2
− u + 1
c
2
u
2
+ u + 1
c
4
, c
7
, c
8
c
9
, c
10
u
2
c
6
, c
11
(u + 1)
2
c
12
(u − 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
y
2
+ y + 1
c
4
, c
7
, c
8
c
9
, c
10
y
2
c
6
, c
11
, c
12
(y − 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0
b = 0.500000 + 0.866025I
c = 0
d = 0.500000 + 0.866025I
1.64493 − 2.02988I 9.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = 0
b = 0.500000 − 0.866025I
c = 0
d = 0.500000 − 0.866025I
1.64493 + 2.02988I 9.00000 − 3.46410I
11
III. I
u
3
= hd + 2u + 1, c + u, b − u, a − u, u
2
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u + 1
a
6
=
u
u
a
7
=
u
u
a
10
=
−u
−2u − 1
a
8
=
0
−u − 1
a
1
=
−u
−u
a
4
=
0
u
a
9
=
0
−u − 1
a
12
=
−u
−u
a
11
=
−u
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u − 1
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
u
2
− u + 1
c
2
u
2
+ u + 1
c
4
, c
6
, c
8
c
11
, c
12
u
2
c
7
(u − 1)
2
c
9
, c
10
(u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
y
2
+ y + 1
c
4
, c
6
, c
8
c
11
, c
12
y
2
c
7
, c
9
, c
10
(y − 1)
2
14
(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
c = 0.500000 − 0.866025I
d = − 1.73205I
−1.64493 − 2.02988I −3.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = −0.500000 − 0.866025I
b = −0.500000 − 0.866025I
c = 0.500000 + 0.866025I
d = 1.73205I
−1.64493 + 2.02988I −3.00000 − 3.46410I
15
IV. I
u
4
= hda + a
2
+ au + a − 1, c + a, b − a − u − 1, u
2
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u + 1
a
6
=
u
u
a
7
=
a
a + u + 1
a
10
=
−a
d
a
8
=
0
d + a + u + 1
a
1
=
−u
−u
a
4
=
0
u
a
9
=
0
d + a + u + 1
a
12
=
a − u
a + 1
a
11
=
a
a + u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −d
2
u + 2a
2
u + a
2
+ 2d + 3u + 5
(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 2.36062 + 3.50810I
17
V. I
v
1
= ha, d − 1, c + a − 1, b + 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
1
0
a
3
=
1
0
a
6
=
1
0
a
7
=
0
−1
a
10
=
1
1
a
8
=
1
0
a
1
=
1
0
a
4
=
1
0
a
9
=
1
0
a
12
=
1
1
a
11
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
8
u
c
6
, c
7
u − 1
c
9
, c
10
, c
11
c
12
u + 1
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
8
y
c
6
, c
7
, c
9
c
10
, c
11
, c
12
y − 1
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −1.00000
c = 1.00000
d = 1.00000
0 0
21
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u(u
2
− u + 1)
2
(u
17
+ 15u
16
+ ··· + 88u − 16)
c
2
u(u
2
+ u + 1)
2
(u
17
+ 5u
16
+ ··· + 11u
2
− 4)
c
3
u(u
2
− u + 1)
2
(u
17
− 14u
16
+ ··· + 6768u − 2592)
c
4
, c
8
u
5
(u
17
− u
16
+ ··· − 1024u − 512)
c
5
u(u
2
− u + 1)
2
(u
17
+ 5u
16
+ ··· + 11u
2
− 4)
c
6
u
2
(u − 1)(u + 1)
2
(u
17
+ 8u
16
+ ··· − 8u − 16)
c
7
u
2
(u − 1)
3
(u
17
− 8u
16
+ ··· − 8u − 16)
c
9
u
2
(u + 1)
3
(u
17
− 8u
16
+ ··· − 8u − 16)
c
10
u
2
(u + 1)
3
(u
17
+ 34u
16
+ ··· + 6176u + 256)
c
11
u
2
(u + 1)
3
(u
17
+ 6u
16
+ ··· + 32u − 256)
c
12
u
2
(u − 1)
2
(u + 1)(u
17
+ 8u
16
+ ··· − 8u − 16)
22
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y(y
2
+ y + 1)
2
(y
17
− 21y
16
+ ··· + 36640y − 256)
c
2
, c
5
y(y
2
+ y + 1)
2
(y
17
+ 15y
16
+ ··· + 88y − 16)
c
3
y(y
2
+ y + 1)
2
(y
17
− 66y
16
+ ··· + 3.67649 × 10
7
y − 6718464)
c
4
, c
8
y
5
(y
17
+ 81y
16
+ ··· − 524288y − 262144)
c
6
, c
12
y
2
(y − 1)
3
(y
17
+ 6y
16
+ ··· + 32y − 256)
c
7
, c
9
y
2
(y − 1)
3
(y
17
− 34y
16
+ ··· + 6176y − 256)
c
10
y
2
(y − 1)
3
(y
17
− 94y
16
+ ··· + 7397888y − 65536)
c
11
y
2
(y − 1)
3
(y
17
+ 66y
16
+ ··· + 2613760y − 65536)
23
