12n
0012
(K12n
0012
)
A knot diagram
1
Linearized knot diagam
3 5 6 9 2 9 11 5 1 12 7 10
Solving Sequence
2,5
3 6
1,10
9 7 4 12 11 8
c
2
c
5
c
1
c
9
c
6
c
4
c
12
c
11
c
7
c
3
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−39u
46
+ 220u
45
+ ··· + 8b − 74, −3u
46
+ 13u
45
+ ··· + 8a − 17, u
47
− 5u
46
+ ··· + 2u − 1i
I
u
2
= hau + b − a, a
4
+ a
3
u − 3a
2
u − 3a
2
+ 2a + u, u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 55 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
= h−39u
46
+ 220u
45
+ · · · + 8b − 74, −3u
46
+ 13u
45
+ · · · + 8a −
17, u
47
− 5u
46
+ · · · + 2u − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
−u
2
a
6
=
u
u
a
1
=
u
2
+ 1
−u
4
a
10
=
3
8
u
46
−
13
8
u
45
+ ··· −
25
4
u +
17
8
39
8
u
46
−
55
2
u
45
+ ··· −
155
8
u +
37
4
a
9
=
−
9
4
u
46
+
39
4
u
45
+ ··· − 5u +
11
4
5
4
u
46
−
89
8
u
45
+ ··· −
105
8
u +
55
8
a
7
=
−u
2
− 1
−
1
8
u
46
+
5
8
u
45
+ ··· +
9
4
u −
1
8
a
4
=
u
4
+ u
2
+ 1
u
4
a
12
=
1
8
u
46
−
1
2
u
45
+ ··· +
7
8
u + 2
1
4
u
46
−
9
8
u
45
+ ··· −
11
8
u +
1
8
a
11
=
7
4
u
46
−
31
4
u
45
+ ··· −
9
2
u +
5
2
−2u
46
+
145
8
u
45
+ ··· +
121
8
u −
93
8
a
8
=
9
4
u
46
−
39
4
u
45
+ ··· + 5u −
11
4
1
2
u
46
+
41
8
u
45
+ ··· +
111
8
u −
67
8
(ii) Obstruction class = −1
(iii) Cusp Shapes =
3
8
u
46
+ 6u
45
+ ··· +
335
8
u −
31
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
47
+ 27u
46
+ ··· − 40u − 1
c
2
, c
5
u
47
+ 5u
46
+ ··· + 2u + 1
c
3
u
47
− 5u
46
+ ··· − 12u + 1
c
4
, c
8
u
47
+ u
46
+ ··· + 640u + 256
c
6
u
47
− 3u
46
+ ··· − 4u + 1
c
7
, c
11
u
47
− 3u
46
+ ··· − 2u + 1
c
9
, c
10
, c
12
u
47
+ 13u
46
+ ··· − 16u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
47
− 9y
46
+ ··· + 512y −1
c
2
, c
5
y
47
+ 27y
46
+ ··· − 40y −1
c
3
y
47
− 45y
46
+ ··· − 152y −1
c
4
, c
8
y
47
− 45y
46
+ ··· + 638976y −65536
c
6
y
47
− 55y
46
+ ··· − 16y −1
c
7
, c
11
y
47
+ 13y
46
+ ··· − 16y −1
c
9
, c
10
, c
12
y
47
+ 45y
46
+ ··· − 8y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.026441 + 0.959622I
a = −0.602286 + 0.667947I
b = −0.28964 + 2.11687I
−1.66847 + 2.07597I −8.19599 − 3.58729I
u = 0.026441 − 0.959622I
a = −0.602286 − 0.667947I
b = −0.28964 − 2.11687I
−1.66847 − 2.07597I −8.19599 + 3.58729I
u = 0.925280 + 0.194352I
a = −1.48851 + 2.82129I
b = 0.86927 − 1.38303I
0.19889 − 8.32605I −2.17091 + 5.11912I
u = 0.925280 − 0.194352I
a = −1.48851 − 2.82129I
b = 0.86927 + 1.38303I
0.19889 + 8.32605I −2.17091 − 5.11912I
u = 0.933825 + 0.065187I
a = 0.120868 + 1.140740I
b = −0.501416 − 0.737782I
−6.55195 − 3.23257I −7.12321 + 3.54877I
u = 0.933825 − 0.065187I
a = 0.120868 − 1.140740I
b = −0.501416 + 0.737782I
−6.55195 + 3.23257I −7.12321 − 3.54877I
u = −0.733010 + 0.802557I
a = −2.03414 − 2.16959I
b = −0.82061 − 2.32750I
4.57748 + 0.17561I −4.00000 + 0.I
u = −0.733010 − 0.802557I
a = −2.03414 + 2.16959I
b = −0.82061 + 2.32750I
4.57748 − 0.17561I −4.00000 + 0.I
u = −0.580519 + 0.922395I
a = 1.018240 − 0.170806I
b = 1.221000 − 0.367778I
−0.75821 − 2.95186I −10.26739 + 0.I
u = −0.580519 − 0.922395I
a = 1.018240 + 0.170806I
b = 1.221000 + 0.367778I
−0.75821 + 2.95186I −10.26739 + 0.I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.886435 + 0.192716I
a = 1.79497 − 2.22924I
b = −0.835270 + 0.957471I
0.91039 − 2.22540I −1.015220 + 0.333993I
u = 0.886435 − 0.192716I
a = 1.79497 + 2.22924I
b = −0.835270 − 0.957471I
0.91039 + 2.22540I −1.015220 − 0.333993I
u = −0.232117 + 1.073970I
a = −0.764740 + 0.107154I
b = −0.930392 − 0.122544I
−3.32956 − 2.69471I −11.16965 + 4.64357I
u = −0.232117 − 1.073970I
a = −0.764740 − 0.107154I
b = −0.930392 + 0.122544I
−3.32956 + 2.69471I −11.16965 − 4.64357I
u = −0.732051 + 0.844151I
a = 2.53321 + 1.61725I
b = 1.64457 + 2.36743I
4.45822 − 5.68770I 0. + 5.85551I
u = −0.732051 − 0.844151I
a = 2.53321 − 1.61725I
b = 1.64457 − 2.36743I
4.45822 + 5.68770I 0. − 5.85551I
u = 0.278391 + 0.834871I
a = 0.195597 − 0.017136I
b = 2.04050 + 2.00587I
6.29656 + 4.54704I −5.32988 − 0.19475I
u = 0.278391 − 0.834871I
a = 0.195597 + 0.017136I
b = 2.04050 − 2.00587I
6.29656 − 4.54704I −5.32988 + 0.19475I
u = −0.468535 + 0.741736I
a = 0.169080 − 0.871234I
b = −0.179453 − 0.614868I
−0.10080 − 1.41741I −3.72542 + 5.86093I
u = −0.468535 − 0.741736I
a = 0.169080 + 0.871234I
b = −0.179453 + 0.614868I
−0.10080 + 1.41741I −3.72542 − 5.86093I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.856224
a = 0.798257
b = 0.0288409
−3.24343 −1.24520
u = 0.277747 + 0.799964I
a = −0.031950 + 0.201535I
b = −1.99709 − 1.62792I
6.39944 − 1.84713I −4.44776 + 5.17555I
u = 0.277747 − 0.799964I
a = −0.031950 − 0.201535I
b = −1.99709 + 1.62792I
6.39944 + 1.84713I −4.44776 − 5.17555I
u = −0.215715 + 0.766929I
a = 0.623937 − 0.580705I
b = −0.040575 − 0.820520I
−0.273553 − 1.319440I −1.87412 + 4.02854I
u = −0.215715 − 0.766929I
a = 0.623937 + 0.580705I
b = −0.040575 + 0.820520I
−0.273553 + 1.319440I −1.87412 − 4.02854I
u = −0.443222 + 1.123720I
a = −0.306235 − 0.216455I
b = −1.89659 − 0.62970I
2.20908 − 1.05804I 0
u = −0.443222 − 1.123720I
a = −0.306235 + 0.216455I
b = −1.89659 + 0.62970I
2.20908 + 1.05804I 0
u = −0.403667 + 1.155110I
a = 0.258478 − 0.232515I
b = 2.08868 − 0.64022I
1.85423 − 6.77428I 0
u = −0.403667 − 1.155110I
a = 0.258478 + 0.232515I
b = 2.08868 + 0.64022I
1.85423 + 6.77428I 0
u = 0.348879 + 1.251500I
a = 0.68603 + 1.45153I
b = 2.92026 + 3.49905I
−3.64593 + 1.84085I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.348879 − 1.251500I
a = 0.68603 − 1.45153I
b = 2.92026 − 3.49905I
−3.64593 − 1.84085I 0
u = 0.467749 + 1.244070I
a = −0.178953 + 0.487783I
b = −0.489450 + 0.885522I
−6.97041 + 4.72417I 0
u = 0.467749 − 1.244070I
a = −0.178953 − 0.487783I
b = −0.489450 − 0.885522I
−6.97041 − 4.72417I 0
u = 0.339549 + 1.291160I
a = −1.09098 − 1.58824I
b = −4.42832 − 3.64735I
−4.58314 − 4.07496I 0
u = 0.339549 − 1.291160I
a = −1.09098 + 1.58824I
b = −4.42832 + 3.64735I
−4.58314 + 4.07496I 0
u = 0.552325 + 1.220220I
a = −1.88083 + 0.66995I
b = −4.90066 + 2.68102I
−2.19262 + 7.48494I 0
u = 0.552325 − 1.220220I
a = −1.88083 − 0.66995I
b = −4.90066 − 2.68102I
−2.19262 − 7.48494I 0
u = −0.643389 + 0.042276I
a = 0.307917 + 0.399006I
b = −0.124452 − 1.032220I
5.17223 − 2.96380I 1.27670 + 2.94526I
u = −0.643389 − 0.042276I
a = 0.307917 − 0.399006I
b = −0.124452 + 1.032220I
5.17223 + 2.96380I 1.27670 − 2.94526I
u = 0.564900 + 1.232880I
a = 2.21676 − 0.36369I
b = 6.02517 − 2.25245I
−2.95715 + 13.73810I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.564900 − 1.232880I
a = 2.21676 + 0.36369I
b = 6.02517 + 2.25245I
−2.95715 − 13.73810I 0
u = 0.435963 + 1.291860I
a = −0.743213 − 0.242847I
b = −2.18896 + 0.57747I
−10.77840 + 1.57100I 0
u = 0.435963 − 1.291860I
a = −0.743213 + 0.242847I
b = −2.18896 − 0.57747I
−10.77840 − 1.57100I 0
u = 0.509079 + 1.270360I
a = 0.692225 + 0.316516I
b = 2.63689 + 1.02049I
−10.24110 + 8.40839I 0
u = 0.509079 − 1.270360I
a = 0.692225 − 0.316516I
b = 2.63689 − 1.02049I
−10.24110 − 8.40839I 0
u = −0.022449 + 0.247534I
a = 1.60540 − 1.10956I
b = −0.337887 − 0.386120I
−0.255033 − 1.107400I −3.72059 + 6.13127I
u = −0.022449 − 0.247534I
a = 1.60540 + 1.10956I
b = −0.337887 + 0.386120I
−0.255033 + 1.107400I −3.72059 − 6.13127I
9
II. I
u
2
= hau + b − a, a
4
+ a
3
u − 3a
2
u − 3a
2
+ 2a + 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
1
=
−u
−u
a
10
=
a
−au + a
a
9
=
0
−au
a
7
=
u
a
2
+ u
a
4
=
0
u
a
12
=
a
2
u + a
2
− u
a
2
u + 2a
2
− u
a
11
=
a
3
u + 2a
2a
3
u + a
3
− au + 2a
a
8
=
0
−au
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2a
3
u − 4a
3
− 5a
2
u + 12au + 17a + 5u − 6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
4
, c
8
u
8
c
6
, c
9
, c
10
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
7
(u
4
− u
3
+ u
2
+ 1)
2
c
11
(u
4
+ u
3
+ u
2
+ 1)
2
c
12
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
4
c
4
, c
8
y
8
c
6
, c
9
, c
10
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
7
, c
11
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −0.241378 − 0.595609I
b = −0.877879 − 0.684374I
−0.211005 − 0.614778I −5.86133 − 2.84273I
u = −0.500000 + 0.866025I
a = 0.636501 − 0.088765I
b = 0.877879 − 0.684374I
−0.21101 − 3.44499I −1.10064 + 8.92228I
u = −0.500000 + 0.866025I
a = −1.29206 − 0.86707I
b = −2.68899 − 0.18165I
6.79074 + 1.13408I 0.90087 + 2.75771I
u = −0.500000 + 0.866025I
a = 1.39694 + 0.68542I
b = 2.68899 − 0.18165I
6.79074 − 5.19385I 1.56110 + 7.61722I
u = −0.500000 − 0.866025I
a = −0.241378 + 0.595609I
b = −0.877879 + 0.684374I
−0.211005 + 0.614778I −5.86133 + 2.84273I
u = −0.500000 − 0.866025I
a = 0.636501 + 0.088765I
b = 0.877879 + 0.684374I
−0.21101 + 3.44499I −1.10064 − 8.92228I
u = −0.500000 − 0.866025I
a = −1.29206 + 0.86707I
b = −2.68899 + 0.18165I
6.79074 − 1.13408I 0.90087 − 2.75771I
u = −0.500000 − 0.866025I
a = 1.39694 − 0.68542I
b = 2.68899 + 0.18165I
6.79074 + 5.19385I 1.56110 − 7.61722I
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
4
)(u
47
+ 27u
46
+ ··· − 40u − 1)
c
2
((u
2
+ u + 1)
4
)(u
47
+ 5u
46
+ ··· + 2u + 1)
c
3
((u
2
− u + 1)
4
)(u
47
− 5u
46
+ ··· − 12u + 1)
c
4
, c
8
u
8
(u
47
+ u
46
+ ··· + 640u + 256)
c
5
((u
2
− u + 1)
4
)(u
47
+ 5u
46
+ ··· + 2u + 1)
c
6
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
47
− 3u
46
+ ··· − 4u + 1)
c
7
((u
4
− u
3
+ u
2
+ 1)
2
)(u
47
− 3u
46
+ ··· − 2u + 1)
c
9
, c
10
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
47
+ 13u
46
+ ··· − 16u − 1)
c
11
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
47
− 3u
46
+ ··· − 2u + 1)
c
12
((u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
)(u
47
+ 13u
46
+ ··· − 16u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
4
)(y
47
− 9y
46
+ ··· + 512y −1)
c
2
, c
5
((y
2
+ y + 1)
4
)(y
47
+ 27y
46
+ ··· − 40y −1)
c
3
((y
2
+ y + 1)
4
)(y
47
− 45y
46
+ ··· − 152y −1)
c
4
, c
8
y
8
(y
47
− 45y
46
+ ··· + 638976y −65536)
c
6
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
47
− 55y
46
+ ··· − 16y −1)
c
7
, c
11
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
47
+ 13y
46
+ ··· − 16y −1)
c
9
, c
10
, c
12
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
47
+ 45y
46
+ ··· − 8y −1)
15
