12n
0056
(K12n
0056
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 12 10 5 11 8 6 11
Solving Sequence
2,5
3
6,8
9 1
4,11
10 12 7
c
2
c
5
c
8
c
1
c
4
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
= h−303u
16
+ 1548u
15
+ ··· + 4864d + 3360, 413u
16
− 1717u
15
+ ··· + 4864c − 1676,
− 306u
16
+ 1521u
15
+ ··· + 2432b + 652, 30u
16
− 53u
15
+ ··· + 1216a − 204,
u
17
− 5u
16
+ ··· − 11u
2
+ 4i
I
u
2
= hd + u, c + u, b − u − 1, a, u
2
+ u + 1i
I
u
3
= hd + u + 1, c, b + u + 1, a, u
2
+ u + 1i
I
u
4
= hd − c + u + 1, cb − 1, a, u
2
+ u + 1i
I
v
1
= hc, d + 1, b, 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
= h−303u
16
+ 1548u
15
+ · · · + 4864d + 3360, 413u
16
− 1717u
15
+ · · · +
4864c − 1676, −306u
16
+ 1521u
15
+ · · · + 2432b + 652, 30u
16
− 53u
15
+ · · · +
1216a − 204, 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
8
=
−0.0246711u
16
+ 0.0435855u
15
+ ··· − 4.48355u + 0.167763
0.125822u
16
− 0.625411u
15
+ ··· + 0.166118u − 0.268092
a
9
=
−0.0246711u
16
+ 0.0435855u
15
+ ··· − 4.48355u + 0.167763
0.164474u
16
− 0.808799u
15
+ ··· + 0.0674342u − 0.587171
a
1
=
u
2
+ 1
−u
4
a
4
=
u
4
+ u
2
+ 1
u
4
a
11
=
−0.0849095u
16
+ 0.353002u
15
+ ··· − 3.22204u + 0.344572
0.0622944u
16
− 0.318257u
15
+ ··· − 0.00246711u − 0.690789
a
10
=
−0.00596217u
16
− 0.0536595u
15
+ ··· − 4.86842u + 0.451480
0.119038u
16
− 0.592722u
15
+ ··· − 0.0246711u − 0.517270
a
12
=
−0.0814145u
16
+ 0.337582u
15
+ ··· − 2.63322u + 0.603618
0.0657895u
16
− 0.333676u
15
+ ··· + 0.586349u − 0.431743
a
7
=
0.118627u
16
− 0.572985u
15
+ ··· + 2.09539u − 0.0238487
−0.0750411u
16
+ 0.305099u
15
+ ··· − 0.0254934u + 0.236842
(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
11
u
17
− 8u
16
+ ··· − 8u − 16
c
7
, c
10
u
17
+ 8u
16
+ ··· − 8u − 16
c
9
u
17
+ 6u
16
+ ··· + 32u − 256
c
12
u
17
+ 34u
16
+ ··· + 6176u + 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
11
y
17
− 34y
16
+ ··· + 6176y − 256
c
7
, c
10
y
17
+ 6y
16
+ ··· + 32y − 256
c
9
y
17
+ 66y
16
+ ··· + 2613760y − 65536
c
12
y
17
− 94y
16
+ ··· + 7397888y − 65536
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.589168 + 0.828507I
a = 0.502465 − 0.319378I
b = 0.252552 − 0.424714I
c = 1.35395 − 1.45051I
d = 1.53322 − 1.04879I
0.79868 − 2.33972I −0.33078 + 5.26516I
u = −0.589168 − 0.828507I
a = 0.502465 + 0.319378I
b = 0.252552 + 0.424714I
c = 1.35395 + 1.45051I
d = 1.53322 + 1.04879I
0.79868 + 2.33972I −0.33078 − 5.26516I
u = −0.403846 + 0.948035I
a = −0.292348 − 0.569503I
b = −0.523078 − 0.308956I
c = −0.142785 − 0.400695I
d = −0.241825 − 1.074000I
−0.77904 − 2.74622I 2.48507 + 7.16740I
u = −0.403846 − 0.948035I
a = −0.292348 + 0.569503I
b = −0.523078 + 0.308956I
c = −0.142785 + 0.400695I
d = −0.241825 + 1.074000I
−0.77904 + 2.74622I 2.48507 − 7.16740I
u = 0.329450 + 1.030540I
a = 0.752669 + 0.404387I
b = 2.49667 − 0.33313I
c = 0.335662 + 0.165758I
d = −0.275871 + 0.445429I
0.72956 + 1.37071I 0.698150 − 0.213889I
u = 0.329450 − 1.030540I
a = 0.752669 − 0.404387I
b = 2.49667 + 0.33313I
c = 0.335662 − 0.165758I
d = −0.275871 − 0.445429I
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 = 0.45151 − 2.07264I
b = 0.686769 − 0.651916I
c = −1.44216 + 0.34761I
d = 0.136010 + 0.385037I
−15.3110 − 5.6503I 2.10303 + 1.68119I
u = 1.349370 − 0.320500I
a = 0.45151 + 2.07264I
b = 0.686769 + 0.651916I
c = −1.44216 − 0.34761I
d = 0.136010 − 0.385037I
−15.3110 + 5.6503I 2.10303 − 1.68119I
u = 0.76686 + 1.31677I
a = −1.58212 + 0.24955I
b = −2.80254 − 0.41679I
c = 0.64759 − 1.27273I
d = 0.83285 − 2.52656I
−18.4182 + 12.9335I 1.01650 − 5.27491I
u = 0.76686 − 1.31677I
a = −1.58212 − 0.24955I
b = −2.80254 + 0.41679I
c = 0.64759 + 1.27273I
d = 0.83285 + 2.52656I
−18.4182 − 12.9335I 1.01650 + 5.27491I
u = 0.249371 + 0.383586I
a = −0.557024 − 1.287010I
b = −0.300121 + 0.720580I
c = −0.04416 − 1.47679I
d = −0.837375 + 0.566407I
−1.75773 + 0.71028I −3.71531 + 0.02644I
u = 0.249371 − 0.383586I
a = −0.557024 + 1.287010I
b = −0.300121 − 0.720580I
c = −0.04416 + 1.47679I
d = −0.837375 − 0.566407I
−1.75773 − 0.71028I −3.71531 − 0.02644I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.275145
a = 1.98253
b = 0.514913
c = 1.10798
d = −0.110369
1.13318 9.61860
u = 0.30683 + 1.77436I
a = 1.87716 + 1.02764I
b = 2.72694 + 1.21615I
c = −0.374205 + 0.884961I
d = 0.39521 + 2.00728I
−9.63429 + 3.26152I −0.10201 − 1.44169I
u = 0.30683 − 1.77436I
a = 1.87716 − 1.02764I
b = 2.72694 − 1.21615I
c = −0.374205 − 0.884961I
d = 0.39521 − 2.00728I
−9.63429 − 3.26152I −0.10201 + 1.44169I
u = 0.62871 + 1.82695I
a = 2.35642 + 0.55040I
b = 3.20534 + 0.91973I
c = 0.112128 − 0.993507I
d = 0.51296 − 2.44755I
17.4865 + 1.7702I 0.036073 − 0.657690I
u = 0.62871 − 1.82695I
a = 2.35642 − 0.55040I
b = 3.20534 − 0.91973I
c = 0.112128 + 0.993507I
d = 0.51296 + 2.44755I
17.4865 − 1.7702I 0.036073 + 0.657690I
7
II. I
u
2
= hd + u, c + u, 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
8
=
0
u + 1
a
9
=
0
u + 1
a
1
=
−u
−u
a
4
=
0
u
a
11
=
−u
−u
a
10
=
−u
1
a
12
=
−u
−u
a
7
=
u
u
(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
6
, c
8
c
11
, c
12
u
2
c
7
, c
9
(u + 1)
2
c
10
(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
6
, c
8
c
11
, c
12
y
2
c
7
, c
9
, c
10
(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 = 0.500000 − 0.866025I
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 = 0.500000 + 0.866025I
d = 0.500000 + 0.866025I
1.64493 + 2.02988I 9.00000 − 3.46410I
11
III. I
u
3
= 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
8
=
0
−u − 1
a
9
=
0
−u − 1
a
1
=
−u
−u
a
4
=
0
u
a
11
=
0
−u − 1
a
10
=
0
−u − 1
a
12
=
−u
−2u − 1
a
7
=
0
−u − 1
(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
7
, c
8
c
9
, c
10
u
2
c
6
(u − 1)
2
c
11
, c
12
(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
7
, c
8
c
9
, c
10
y
2
c
6
, c
11
, c
12
(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
b = −0.500000 − 0.866025I
c = 0
d = −0.500000 − 0.866025I
−1.64493 − 2.02988I −3.00000 + 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 −3.00000 − 3.46410I
15
IV. I
u
4
= hd − c + u + 1, cb − 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
8
=
0
b
a
9
=
0
b
a
1
=
−u
−u
a
4
=
0
u
a
11
=
c
c − u − 1
a
10
=
c
c + b − u − 1
a
12
=
c − u
c − 2u − 1
a
7
=
c
c − u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = c
2
u − b
2
u + c
2
+ 4u + 4
(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 0.58899 + 3.27641I
17
V. I
v
1
= hc, d + 1, b, a − 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
1
0
a
3
=
1
0
a
6
=
1
0
a
8
=
1
0
a
9
=
1
0
a
1
=
1
0
a
4
=
1
0
a
11
=
0
−1
a
10
=
1
−1
a
12
=
1
−1
a
7
=
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
, c
9
c
12
u + 1
c
10
, c
11
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 = 1.00000
b = 0
c = 0
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)
2
(u + 1)(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
+ 6u
16
+ ··· + 32u − 256)
c
10
u
2
(u − 1)
3
(u
17
+ 8u
16
+ ··· − 8u − 16)
c
11
u
2
(u − 1)(u + 1)
2
(u
17
− 8u
16
+ ··· − 8u − 16)
c
12
u
2
(u + 1)
3
(u
17
+ 34u
16
+ ··· + 6176u + 256)
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
11
y
2
(y − 1)
3
(y
17
− 34y
16
+ ··· + 6176y − 256)
c
7
, c
10
y
2
(y − 1)
3
(y
17
+ 6y
16
+ ··· + 32y − 256)
c
9
y
2
(y − 1)
3
(y
17
+ 66y
16
+ ··· + 2613760y − 65536)
c
12
y
2
(y − 1)
3
(y
17
− 94y
16
+ ··· + 7397888y − 65536)
23
