12n
0220
(K12n
0220
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 12 9 3 6 8 5 6 11
Solving Sequence
5,12 3,6
2 1
4,8
7 11 10 9
c
5
c
2
c
1
c
4
c
7
c
11
c
10
c
9
c
3
, c
6
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h2281u
12
+ 1307u
11
+ ··· + 44956d + 11490, 1947u
12
+ 1293u
11
+ ··· + 22478c + 16277,
573u
12
+ 2043u
11
+ ··· + 44956b + 16722, −1947u
12
− 1293u
11
+ ··· + 22478a − 16277,
u
13
+ u
12
+ 2u
11
+ u
10
+ 5u
9
+ u
8
+ 4u
7
+ u
6
+ 15u
5
+ 5u
4
+ 16u
3
+ 5u
2
+ 12u + 4i
I
u
2
= h−u
4
+ u
2
a − 2u
3
+ au − u
2
+ d + 2u + 2, u
4
+ 3u
3
+ 5u
2
+ c − a + 3u + 1, u
4
+ 2u
3
− au + 2u
2
+ b,
− u
4
a − 3u
3
a + 2u
4
− 5u
2
a + 4u
3
+ a
2
− 3au + 3u
2
− a − 2u − 1, u
5
+ 2u
4
+ 2u
3
+ u + 1i
I
u
3
= hd, c + u, b, a − 1, u
2
− u + 1i
I
u
4
= hd − u − 1, c − 1, b + 1, a − u, u
2
+ u + 1i
I
u
5
= h−cu + d − c + 1, ca − cu + au, b + 1, u
2
+ u + 1i
I
v
1
= ha, d − 1, c + a, b + 1, v + 1i
* 5 irreducible components of dim
C
= 0, with total 28 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
= h2281u
12
+ 1307u
11
+ · · · + 4.50 × 10
4
d + 1.15 × 10
4
, 1947u
12
+ 1293u
11
+
· · · + 2.25 × 10
4
c + 1.63 × 10
4
, 573u
12
+ 2043u
11
+ · · · + 4.50 × 10
4
b + 1.67 ×
10
4
, −1947u
12
−1293u
11
+· · ·+2.25×10
4
a−1.63×10
4
, u
13
+u
12
+· · ·+12u+4i
(i) Arc colorings
a
5
=
1
0
a
12
=
0
u
a
3
=
0.0866180u
12
+ 0.0575229u
11
+ ··· + 0.482205u + 0.724130
−0.0127458u
12
− 0.0454444u
11
+ ··· + 0.294221u − 0.371964
a
6
=
1
−u
2
a
2
=
0.0738722u
12
+ 0.0120785u
11
+ ··· + 0.776426u + 0.352167
−0.0127458u
12
− 0.0454444u
11
+ ··· + 0.294221u − 0.371964
a
1
=
−u
3
u
5
+ u
3
+ u
a
4
=
0.0683446u
12
− 0.0285724u
11
+ ··· −0.389169u + 0.832214
−0.00298069u
12
− 0.0420411u
11
+ ··· − 0.370985u − 0.374944
a
8
=
−0.0866180u
12
− 0.0575229u
11
+ ··· −0.482205u − 0.724130
−0.0507385u
12
− 0.0290729u
11
+ ··· + 0.296890u − 0.255583
a
7
=
−0.0683446u
12
+ 0.0285724u
11
+ ··· + 0.389169u − 0.832214
0.0180176u
12
− 0.119005u
11
+ ··· + 0.518640u + 0.0127236
a
11
=
−u
u
3
+ u
a
10
=
u
3
u
3
+ u
a
9
=
−0.0738722u
12
− 0.0120785u
11
+ ··· −0.776426u − 0.352167
−0.0613266u
12
+ 0.0902438u
11
+ ··· + 0.740257u − 0.124789
(ii) Obstruction class = −1
(iii) Cusp Shapes =
2015
11239
u
12
−
4290
11239
u
11
+ ··· +
6386
11239
u −
108192
11239
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
9
u
13
− u
12
+ ··· + 16u + 1
c
2
, c
4
, c
6
c
8
u
13
− 5u
12
+ ··· − 4u + 1
c
3
, c
7
u
13
− 3u
12
+ ··· − 32u + 32
c
5
, c
11
u
13
+ u
12
+ ··· + 12u + 4
c
10
u
13
− u
12
+ ··· + 1508u + 548
c
12
u
13
+ 3u
12
+ ··· + 104u − 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
9
y
13
+ 25y
12
+ ··· −260y − 1
c
2
, c
4
, c
6
c
8
y
13
+ y
12
+ ··· + 16y − 1
c
3
, c
7
y
13
+ 15y
12
+ ··· + 15616y
2
− 1024
c
5
, c
11
y
13
+ 3y
12
+ ··· + 104y − 16
c
10
y
13
+ 27y
12
+ ··· + 1970472y − 300304
c
12
y
13
+ 15y
12
+ ··· + 21024y − 256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.386403 + 0.917053I
a = −0.849710 + 0.767631I
b = −0.801603 − 0.173700I
c = 0.849710 − 0.767631I
d = 0.330147 − 0.102461I
−2.92013 − 2.62586I −15.8235 + 5.3570I
u = −0.386403 − 0.917053I
a = −0.849710 − 0.767631I
b = −0.801603 + 0.173700I
c = 0.849710 + 0.767631I
d = 0.330147 + 0.102461I
−2.92013 + 2.62586I −15.8235 − 5.3570I
u = 0.416573 + 0.881458I
a = 0.686659 + 0.124521I
b = 0.221947 + 0.150698I
c = −0.686659 − 0.124521I
d = −0.283854 + 0.579828I
−0.33676 + 1.74909I −2.22256 − 3.20069I
u = 0.416573 − 0.881458I
a = 0.686659 − 0.124521I
b = 0.221947 − 0.150698I
c = −0.686659 + 0.124521I
d = −0.283854 − 0.579828I
−0.33676 − 1.74909I −2.22256 + 3.20069I
u = 1.124080 + 0.602862I
a = −0.176205 − 1.075190I
b = −0.16802 + 1.50582I
c = 0.176205 + 1.075190I
d = 1.130610 + 0.299207I
4.55733 + 1.91344I −6.23694 − 1.74226I
u = 1.124080 − 0.602862I
a = −0.176205 + 1.075190I
b = −0.16802 − 1.50582I
c = 0.176205 − 1.075190I
d = 1.130610 − 0.299207I
4.55733 − 1.91344I −6.23694 + 1.74226I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.543511 + 1.275200I
a = 1.113650 + 0.332769I
b = 0.536277 − 1.193890I
c = −1.113650 − 0.332769I
d = −1.406970 − 0.093004I
1.88235 + 4.50009I −8.08386 − 3.64476I
u = 0.543511 − 1.275200I
a = 1.113650 − 0.332769I
b = 0.536277 + 1.193890I
c = −1.113650 + 0.332769I
d = −1.406970 + 0.093004I
1.88235 − 4.50009I −8.08386 + 3.64476I
u = −1.173290 + 0.753740I
a = −0.464126 + 0.518194I
b = 1.48175 − 1.16585I
c = 0.464126 − 0.518194I
d = 2.02304 + 0.07401I
13.3607 + 6.1261I −8.08998 − 1.87384I
u = −1.173290 − 0.753740I
a = −0.464126 − 0.518194I
b = 1.48175 + 1.16585I
c = 0.464126 + 0.518194I
d = 2.02304 − 0.07401I
13.3607 − 6.1261I −8.08998 + 1.87384I
u = −0.85913 + 1.17284I
a = 0.84945 − 1.49776I
b = 1.47195 + 0.93931I
c = −0.84945 + 1.49776I
d = −2.08790 + 0.18218I
11.8885 − 13.4346I −9.57192 + 6.10692I
u = −0.85913 − 1.17284I
a = 0.84945 + 1.49776I
b = 1.47195 − 0.93931I
c = −0.84945 − 1.49776I
d = −2.08790 − 0.18218I
11.8885 + 13.4346I −9.57192 − 6.10692I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.330680
a = 0.680555
b = −0.484585
c = −0.680555
d = −0.410167
−0.936151 −9.94250
7
II. I
u
2
= h−u
4
− 2u
3
+ · · · + d + 2, u
4
+ 3u
3
+ · · · − a + 1, u
4
+ 2u
3
− au +
2u
2
+ b, −u
4
a + 2u
4
+ · · · − a − 1, u
5
+ 2u
4
+ 2u
3
+ u + 1i
(i) Arc colorings
a
5
=
1
0
a
12
=
0
u
a
3
=
a
−u
4
− 2u
3
+ au − 2u
2
a
6
=
1
−u
2
a
2
=
−u
4
− 2u
3
+ au − 2u
2
+ a
−u
4
− 2u
3
+ au − 2u
2
a
1
=
−u
3
−2u
4
− u
3
− 1
a
4
=
−u
4
a − 2u
3
a − u
2
a − u
2
− 3u − 1
−u
4
a − u
3
a + u
3
+ u
2
− a
a
8
=
−u
4
− 3u
3
− 5u
2
+ a − 3u − 1
u
4
− u
2
a + 2u
3
− au + u
2
− 2u − 2
a
7
=
−u
4
a − 2u
3
a − u
4
− u
2
a − 3u
3
− 5u
2
− 3u − 1
u
3
a + u
4
− u
2
a + u
3
− au + a − 3u − 2
a
11
=
−u
u
3
+ u
a
10
=
u
3
u
3
+ u
a
9
=
−u
4
− 4u
3
+ au − 6u
2
+ a − 3u
−u
3
a − u
2
a − au − 3u − 3
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
4
+ u
3
− 2u
2
− 5u − 10
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
9
u
10
− u
9
+ ··· + 800u + 256
c
2
, c
4
, c
6
c
8
u
10
− 3u
9
+ 5u
8
+ 3u
7
− 12u
6
+ 10u
5
+ 17u
4
− 18u
3
− 23u
2
+ 8u + 16
c
3
, c
7
(u
5
+ u
4
+ 8u
3
+ u
2
− 4u + 4)
2
c
5
, c
11
(u
5
+ 2u
4
+ 2u
3
+ u + 1)
2
c
10
(u
5
− 2u
4
+ 14u
3
+ 16u
2
+ 9u + 9)
2
c
12
(u
5
+ 6u
3
+ u − 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
9
y
10
+ 37y
9
+ ··· + 56832y + 65536
c
2
, c
4
, c
6
c
8
y
10
+ y
9
+ ··· − 800y + 256
c
3
, c
7
(y
5
+ 15y
4
+ 54y
3
− 73y
2
+ 8y − 16)
2
c
5
, c
11
(y
5
+ 6y
3
+ y − 1)
2
c
10
(y
5
+ 24y
4
+ 278y
3
+ 32y
2
− 207y − 81)
2
c
12
(y
5
+ 12y
4
+ 38y
3
+ 12y
2
+ y − 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.436447 + 0.655029I
a = 0.445445 + 1.296420I
b = 1.049680 − 0.199668I
c = 1.03494 − 3.53452I
d = −2.83647 − 1.62756I
−3.34738 + 1.37362I −12.45374 − 4.59823I
u = 0.436447 + 0.655029I
a = −1.03494 + 3.53452I
b = −1.062450 − 0.192555I
c = −0.445445 − 1.296420I
d = 0.202150 − 0.254271I
−3.34738 + 1.37362I −12.45374 − 4.59823I
u = 0.436447 − 0.655029I
a = 0.445445 − 1.296420I
b = 1.049680 + 0.199668I
c = 1.03494 + 3.53452I
d = −2.83647 + 1.62756I
−3.34738 − 1.37362I −12.45374 + 4.59823I
u = 0.436447 − 0.655029I
a = −1.03494 − 3.53452I
b = −1.062450 + 0.192555I
c = −0.445445 + 1.296420I
d = 0.202150 + 0.254271I
−3.34738 − 1.37362I −12.45374 + 4.59823I
u = −0.668466
a = 0.266201 + 0.900637I
b = −0.673909 − 0.602045I
c = −0.266201 + 0.900637I
d = −0.554957 + 0.199598I
−0.737094 −7.65040
u = −0.668466
a = 0.266201 − 0.900637I
b = −0.673909 + 0.602045I
c = −0.266201 − 0.900637I
d = −0.554957 − 0.199598I
−0.737094 −7.65040
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.10221 + 1.09532I
a = −0.730929 + 0.410318I
b = 0.89973 − 1.70648I
c = −0.554227 + 1.236440I
d = −1.69011 + 0.15931I
14.4080 − 4.0569I −7.72106 + 1.95729I
u = −1.10221 + 1.09532I
a = 0.554227 − 1.236440I
b = 1.28694 + 1.51626I
c = 0.730929 − 0.410318I
d = 1.87939 + 0.06460I
14.4080 − 4.0569I −7.72106 + 1.95729I
u = −1.10221 − 1.09532I
a = −0.730929 − 0.410318I
b = 0.89973 + 1.70648I
c = −0.554227 − 1.236440I
d = −1.69011 − 0.15931I
14.4080 + 4.0569I −7.72106 − 1.95729I
u = −1.10221 − 1.09532I
a = 0.554227 + 1.236440I
b = 1.28694 − 1.51626I
c = 0.730929 + 0.410318I
d = 1.87939 − 0.06460I
14.4080 + 4.0569I −7.72106 − 1.95729I
12
III. I
u
3
= hd, c + u, b, a − 1, u
2
− u + 1i
(i) Arc colorings
a
5
=
1
0
a
12
=
0
u
a
3
=
1
0
a
6
=
1
−u + 1
a
2
=
1
0
a
1
=
1
0
a
4
=
1
0
a
8
=
−u
0
a
7
=
−u
0
a
11
=
−u
u − 1
a
10
=
−1
u − 1
a
9
=
−u − 1
u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u − 7
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
u
2
c
5
, c
10
u
2
− u + 1
c
6
(u − 1)
2
c
8
, c
9
(u + 1)
2
c
11
, c
12
u
2
+ u + 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
y
2
c
5
, c
10
, c
11
c
12
y
2
+ y + 1
c
6
, c
8
, c
9
(y − 1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 1.00000
b = 0
c = −0.500000 − 0.866025I
d = 0
−1.64493 + 2.02988I −9.00000 − 3.46410I
u = 0.500000 − 0.866025I
a = 1.00000
b = 0
c = −0.500000 + 0.866025I
d = 0
−1.64493 − 2.02988I −9.00000 + 3.46410I
16
IV. I
u
4
= hd − u − 1, c − 1, b + 1, a − u, u
2
+ u + 1i
(i) Arc colorings
a
5
=
1
0
a
12
=
0
u
a
3
=
u
−1
a
6
=
1
u + 1
a
2
=
u − 1
−1
a
1
=
−1
0
a
4
=
u
−1
a
8
=
1
u + 1
a
7
=
1
u + 1
a
11
=
−u
u + 1
a
10
=
1
u + 1
a
9
=
1
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u − 7
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
6
, c
7
c
8
, c
9
u
2
c
4
(u + 1)
2
c
5
, c
10
, c
12
u
2
+ u + 1
c
11
u
2
− u + 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
2
c
3
, c
6
, c
7
c
8
, c
9
y
2
c
5
, c
10
, c
11
c
12
y
2
+ y + 1
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −0.500000 + 0.866025I
b = −1.00000
c = 1.00000
d = 0.500000 + 0.866025I
−1.64493 − 2.02988I −9.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = −0.500000 − 0.866025I
b = −1.00000
c = 1.00000
d = 0.500000 − 0.866025I
−1.64493 + 2.02988I −9.00000 − 3.46410I
20
V. I
u
5
= h−cu + d − c + 1, ca − cu + au, b + 1, u
2
+ u + 1i
(i) Arc colorings
a
5
=
1
0
a
12
=
0
u
a
3
=
a
−1
a
6
=
1
u + 1
a
2
=
a − 1
−1
a
1
=
−1
0
a
4
=
a
−1
a
8
=
c
cu + c − 1
a
7
=
c
cu + c − 1
a
11
=
−u
u + 1
a
10
=
1
u + 1
a
9
=
c + 1
cu + c + u
(ii) Obstruction class = −1
(iii) Cusp Shapes = c
2
u + a
2
u − 2cu + 2au − 2c + 2a + 4u − 10
(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.
21
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
c = ···
d = ···
−3.28987 + 2.02988I −13.6251 − 6.4182I
22
VI. I
v
1
= ha, d − 1, c + a, b + 1, v + 1i
(i) Arc colorings
a
5
=
1
0
a
12
=
−1
0
a
3
=
0
−1
a
6
=
1
0
a
2
=
−1
−1
a
1
=
−1
0
a
4
=
0
−1
a
8
=
0
1
a
7
=
0
1
a
11
=
−1
0
a
10
=
−1
0
a
9
=
−1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
23
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u − 1
c
3
, c
5
, c
7
c
10
, c
11
, c
12
u
c
4
, c
8
, c
9
u + 1
24
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
6
, c
8
, c
9
y − 1
c
3
, c
5
, c
7
c
10
, c
11
, c
12
y
25
(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 = 0
d = 1.00000
−3.28987 −12.0000
26
VII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
2
(u − 1)
3
(u
10
− u
9
+ ··· + 800u + 256)(u
13
− u
12
+ ··· + 16u + 1)
c
2
, c
6
u
2
(u − 1)
3
· (u
10
− 3u
9
+ 5u
8
+ 3u
7
− 12u
6
+ 10u
5
+ 17u
4
− 18u
3
− 23u
2
+ 8u + 16)
· (u
13
− 5u
12
+ ··· − 4u + 1)
c
3
, c
7
u
5
(u
5
+ u
4
+ ··· − 4u + 4)
2
(u
13
− 3u
12
+ ··· − 32u + 32)
c
4
, c
8
u
2
(u + 1)
3
· (u
10
− 3u
9
+ 5u
8
+ 3u
7
− 12u
6
+ 10u
5
+ 17u
4
− 18u
3
− 23u
2
+ 8u + 16)
· (u
13
− 5u
12
+ ··· − 4u + 1)
c
5
, c
11
u(u
2
− u + 1)(u
2
+ u + 1)(u
5
+ 2u
4
+ 2u
3
+ u + 1)
2
· (u
13
+ u
12
+ ··· + 12u + 4)
c
9
u
2
(u + 1)
3
(u
10
− u
9
+ ··· + 800u + 256)(u
13
− u
12
+ ··· + 16u + 1)
c
10
u(u
2
− u + 1)(u
2
+ u + 1)(u
5
− 2u
4
+ 14u
3
+ 16u
2
+ 9u + 9)
2
· (u
13
− u
12
+ ··· + 1508u + 548)
c
12
u(u
2
+ u + 1)
2
(u
5
+ 6u
3
+ u − 1)
2
(u
13
+ 3u
12
+ ··· + 104u − 16)
27
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
9
y
2
(y − 1)
3
(y
10
+ 37y
9
+ ··· + 56832y + 65536)
· (y
13
+ 25y
12
+ ··· −260y − 1)
c
2
, c
4
, c
6
c
8
y
2
(y − 1)
3
(y
10
+ y
9
+ ··· − 800y + 256)(y
13
+ y
12
+ ··· + 16y − 1)
c
3
, c
7
y
5
(y
5
+ 15y
4
+ 54y
3
− 73y
2
+ 8y − 16)
2
· (y
13
+ 15y
12
+ ··· + 15616y
2
− 1024)
c
5
, c
11
y(y
2
+ y + 1)
2
(y
5
+ 6y
3
+ y − 1)
2
(y
13
+ 3y
12
+ ··· + 104y − 16)
c
10
y(y
2
+ y + 1)
2
(y
5
+ 24y
4
+ 278y
3
+ 32y
2
− 207y − 81)
2
· (y
13
+ 27y
12
+ ··· + 1970472y − 300304)
c
12
y(y
2
+ y + 1)
2
(y
5
+ 12y
4
+ 38y
3
+ 12y
2
+ y − 1)
2
· (y
13
+ 15y
12
+ ··· + 21024y − 256)
28
