12n
0817
(K12n
0817
)
A knot diagram
1
Linearized knot diagam
4 6 11 7 10 2 12 11 5 3 7 8
Solving Sequence
7,11
12 8 9
1,5
4 2 3 6 10
c
11
c
7
c
8
c
12
c
4
c
1
c
3
c
6
c
10
c
2
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−139u
18
+ 1059u
17
+ ··· + 4b − 764, −87u
18
+ 677u
17
+ ··· + 8a − 508, u
19
− 9u
18
+ ··· + 12u − 8i
I
u
2
= h2u
12
+ 2u
11
− 15u
10
− 12u
9
+ 43u
8
+ 27u
7
− 53u
6
− 26u
5
+ 20u
4
+ 10u
3
+ 6u
2
+ b − 4u − 2,
u
10
+ u
9
− 7u
8
− 6u
7
+ 18u
6
+ 13u
5
− 18u
4
− 10u
3
+ 2u
2
+ a + 4,
u
13
+ 2u
12
− 7u
11
− 14u
10
+ 19u
9
+ 38u
8
− 22u
7
− 46u
6
+ 6u
5
+ 20u
4
+ 7u
3
+ u
2
− 5u − 1i
I
u
3
= h5a
5
u
2
− 7a
4
u
2
+ ··· − 11a + 26, 2a
5
u
2
+ a
4
u
2
+ ··· + 95a + 46, u
3
+ u
2
− 2u − 1i
* 3 irreducible components of dim
C
= 0, with total 50 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−139u
18
+ 1059u
17
+ · · · + 4b − 764, −87u
18
+ 677u
17
+ · · · + 8a −
508, u
19
− 9u
18
+ · · · + 12u − 8i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
8
=
u
−u
3
+ u
a
9
=
−u
3
+ 2u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
5
=
10.8750u
18
− 84.6250u
17
+ ··· − 51.7500u + 63.5000
139
4
u
18
−
1059
4
u
17
+ ··· − 159u + 191
a
4
=
10.8750u
18
− 84.6250u
17
+ ··· − 51.7500u + 63.5000
87
4
u
18
−
611
4
u
17
+ ··· − 87u + 85
a
2
=
−
63
8
u
18
+
465
8
u
17
+ ··· +
135
4
u − 38
−
1
4
u
18
+
7
4
u
17
+ ··· +
1
2
u − 1
a
3
=
32.6250u
18
− 237.375u
17
+ ··· − 138.750u + 148.500
87
4
u
18
−
611
4
u
17
+ ··· − 87u + 85
a
6
=
169
8
u
18
−
1229
8
u
17
+ ··· −
181
2
u + 96
6u
18
−
85
2
u
17
+ ··· −
47
2
u + 25
a
10
=
1
8
u
18
−
7
8
u
17
+ ··· +
3
4
u + 1
1
4
u
18
−
7
4
u
17
+ ··· −
1
2
u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −25u
18
+ 174u
17
−365u
16
−25u
15
+ 771u
14
+ 163u
13
−1928u
12
+ 630u
11
+ 1143u
10
+
293u
9
− 304u
8
− 532u
7
− 1600u
6
+ 1277u
5
+ 1032u
4
− 324u
3
− 389u
2
+ 90u − 94
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
19
+ u
18
+ ··· + 12u + 1
c
2
, c
6
u
19
− 8u
18
+ ··· − 52u + 8
c
3
, c
5
, c
9
c
10
u
19
− u
18
+ ··· − 2u − 1
c
7
, c
11
, c
12
u
19
− 9u
18
+ ··· + 12u − 8
c
8
u
19
+ 27u
18
+ ··· + 27116u + 3512
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
19
+ 35y
18
+ ··· + 42y −1
c
2
, c
6
y
19
+ 16y
18
+ ··· − 48y −64
c
3
, c
5
, c
9
c
10
y
19
− 11y
18
+ ··· + 10y −1
c
7
, c
11
, c
12
y
19
− 23y
18
+ ··· − 432y −64
c
8
y
19
− 43y
18
+ ··· + 4640976y −12334144
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.091188 + 0.966333I
a = 0.733877 − 0.319784I
b = 0.211486 − 0.489342I
−6.48169 + 2.66282I −2.52143 − 2.56509I
u = −0.091188 − 0.966333I
a = 0.733877 + 0.319784I
b = 0.211486 + 0.489342I
−6.48169 − 2.66282I −2.52143 + 2.56509I
u = −0.904144 + 0.517931I
a = 0.706601 − 1.068440I
b = 0.218968 − 0.639966I
2.53064 − 4.01875I 1.69485 + 6.55873I
u = −0.904144 − 0.517931I
a = 0.706601 + 1.068440I
b = 0.218968 + 0.639966I
2.53064 + 4.01875I 1.69485 − 6.55873I
u = 1.19835
a = 0.103232
b = 0.623676
2.53576 4.52770
u = −0.634175 + 0.351711I
a = −1.29575 + 1.15797I
b = −0.392189 + 0.643839I
0.91268 + 1.32361I −5.06829 + 3.42198I
u = −0.634175 − 0.351711I
a = −1.29575 − 1.15797I
b = −0.392189 − 0.643839I
0.91268 − 1.32361I −5.06829 − 3.42198I
u = −1.060850 + 0.713165I
a = −0.549004 + 0.898836I
b = −0.176307 + 0.591660I
−3.53037 − 8.31890I −1.40883 + 6.39242I
u = −1.060850 − 0.713165I
a = −0.549004 − 0.898836I
b = −0.176307 − 0.591660I
−3.53037 + 8.31890I −1.40883 − 6.39242I
u = 1.289480 + 0.424662I
a = 0.114699 − 0.319331I
b = −0.272999 − 0.853707I
−2.23583 + 2.25035I −0.46692 − 2.77886I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.289480 − 0.424662I
a = 0.114699 + 0.319331I
b = −0.272999 + 0.853707I
−2.23583 − 2.25035I −0.46692 + 2.77886I
u = 0.103961 + 0.369644I
a = −0.936993 − 0.305312I
b = −0.163488 + 0.354518I
−0.056986 + 0.926042I −1.11794 − 7.21094I
u = 0.103961 − 0.369644I
a = −0.936993 + 0.305312I
b = −0.163488 − 0.354518I
−0.056986 − 0.926042I −1.11794 + 7.21094I
u = 1.67359 + 0.17987I
a = 0.094930 + 1.113270I
b = 0.63844 + 2.62086I
9.08712 + 1.06966I −0.590577 − 0.693151I
u = 1.67359 − 0.17987I
a = 0.094930 − 1.113270I
b = 0.63844 − 2.62086I
9.08712 − 1.06966I −0.590577 + 0.693151I
u = 1.73838 + 0.18173I
a = 0.101477 − 1.233980I
b = −0.15677 − 2.84161I
11.82410 + 7.02626I 2.32832 − 4.17154I
u = 1.73838 − 0.18173I
a = 0.101477 + 1.233980I
b = −0.15677 + 2.84161I
11.82410 − 7.02626I 2.32832 + 4.17154I
u = 1.78576 + 0.20016I
a = −0.271457 + 1.238060I
b = −0.21898 + 2.80030I
6.42171 + 12.16180I −0.61302 − 5.43087I
u = 1.78576 − 0.20016I
a = −0.271457 − 1.238060I
b = −0.21898 − 2.80030I
6.42171 − 12.16180I −0.61302 + 5.43087I
6
II.
I
u
2
= h2u
12
+ 2u
11
+ · · · + b − 2, u
10
+ u
9
+ · · · + a + 4, u
13
+ 2u
12
+ · · · − 5u − 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
8
=
u
−u
3
+ u
a
9
=
−u
3
+ 2u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
5
=
−u
10
− u
9
+ 7u
8
+ 6u
7
− 18u
6
− 13u
5
+ 18u
4
+ 10u
3
− 2u
2
− 4
−2u
12
− 2u
11
+ ··· + 4u + 2
a
4
=
−u
10
− u
9
+ 7u
8
+ 6u
7
− 18u
6
− 13u
5
+ 18u
4
+ 10u
3
− 2u
2
− 4
−u
12
− u
11
+ ··· + 4u + 2
a
2
=
−u
10
− u
9
+ 7u
8
+ 6u
7
− 18u
6
− 14u
5
+ 18u
4
+ 14u
3
− 3u
2
− 4u − 3
u
4
− 3u
2
+ 1
a
3
=
−u
12
− u
11
+ 7u
10
+ 5u
9
− 18u
8
− 8u
7
+ 17u
6
+ 3u
5
− 4u
2
+ 4u − 2
−u
12
− u
11
+ ··· + 4u + 2
a
6
=
−u
11
− u
10
+ 7u
9
+ 5u
8
− 18u
7
− 8u
6
+ 17u
5
+ 2u
4
+ 4u
2
− 5u
u
7
− 5u
5
+ 7u
3
+ u
2
− 2u − 1
a
10
=
−u
12
− 2u
11
+ ··· + 2u + 5
−u
3
+ 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes
= −3u
11
− 7u
10
+ 17u
9
+ 43u
8
− 33u
7
− 95u
6
+ 13u
5
+ 73u
4
+ 23u
3
+ 6u
2
− 14u − 14
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
13
+ u
12
+ ··· + 3u
2
− 1
c
2
u
13
− u
12
+ ··· + 2u + 1
c
3
, c
9
u
13
+ u
12
− 4u
11
− 4u
10
+ 6u
9
+ 5u
8
− 2u
7
+ 4u
6
− 2u
5
− 14u
4
+ 7u
2
+ 1
c
5
, c
10
u
13
− u
12
− 4u
11
+ 4u
10
+ 6u
9
− 5u
8
− 2u
7
− 4u
6
− 2u
5
+ 14u
4
− 7u
2
− 1
c
6
u
13
+ u
12
+ ··· + 2u − 1
c
7
u
13
− 2u
12
+ ··· − 5u + 1
c
8
u
13
+ 6u
12
+ ··· + u + 1
c
11
, c
12
u
13
+ 2u
12
+ ··· − 5u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
13
+ 5y
12
+ ··· + 6y −1
c
2
, c
6
y
13
+ 13y
12
+ ··· + 18y −1
c
3
, c
5
, c
9
c
10
y
13
− 9y
12
+ ··· − 14y −1
c
7
, c
11
, c
12
y
13
− 18y
12
+ ··· + 27y −1
c
8
y
13
− 26y
12
+ ··· + 43y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.820152 + 0.104400I
a = 0.753200 + 1.119880I
b = 0.305571 + 0.530288I
1.34031 − 1.84544I 2.80039 + 5.64574I
u = 0.820152 − 0.104400I
a = 0.753200 − 1.119880I
b = 0.305571 − 0.530288I
1.34031 + 1.84544I 2.80039 − 5.64574I
u = −1.26705
a = −0.390134
b = −1.85106
−0.232440 0.645970
u = −1.300200 + 0.250560I
a = 0.343272 − 0.374017I
b = 1.63014 − 0.54583I
−4.74439 − 5.27325I −1.78132 + 3.84852I
u = −1.300200 − 0.250560I
a = 0.343272 + 0.374017I
b = 1.63014 + 0.54583I
−4.74439 + 5.27325I −1.78132 − 3.84852I
u = 1.35503
a = 0.699592
b = 0.402528
1.54960 −4.46630
u = −0.162080 + 0.555123I
a = −0.32922 + 1.58092I
b = −0.924848 + 0.088801I
−8.52981 + 2.40351I −9.03747 − 0.70595I
u = −0.162080 − 0.555123I
a = −0.32922 − 1.58092I
b = −0.924848 − 0.088801I
−8.52981 − 2.40351I −9.03747 + 0.70595I
u = 1.48038 + 0.34329I
a = −0.744111 + 0.018385I
b = −0.450757 − 0.029628I
−2.98672 + 1.03297I −3.12657 + 1.51158I
u = 1.48038 − 0.34329I
a = −0.744111 − 0.018385I
b = −0.450757 + 0.029628I
−2.98672 − 1.03297I −3.12657 − 1.51158I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.78337 + 0.04448I
a = −0.115269 + 1.138540I
b = −0.34421 + 2.57894I
11.19350 + 0.95284I 1.091372 − 0.413941I
u = −1.78337 − 0.04448I
a = −0.115269 − 1.138540I
b = −0.34421 − 2.57894I
11.19350 − 0.95284I 1.091372 + 0.413941I
u = −0.197734
a = −4.12519
b = 1.01673
−3.73260 −11.0720
11
III. I
u
3
=
h5a
5
u
2
−7a
4
u
2
+· · ·−11a +26, 2a
5
u
2
+a
4
u
2
+· · ·+95a +46, u
3
+u
2
−2u−1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
8
=
u
u
2
− u − 1
a
9
=
u
2
− 1
u
2
− u − 1
a
1
=
−u
2
+ 1
u
2
− u − 1
a
5
=
a
−0.178571a
5
u
2
+ 0.250000a
4
u
2
+ ··· + 0.392857a − 0.928571
a
4
=
a
−0.178571a
5
u
2
+ 0.250000a
4
u
2
+ ··· + 0.392857a − 0.928571
a
2
=
−0.214286a
5
u
2
− 0.250000a
4
u
2
+ ··· + 0.0714286a + 1.28571
15
28
a
5
u
2
+
1
2
a
4
u
2
+ ··· +
1
14
a +
2
7
a
3
=
−0.178571a
5
u
2
+ 0.250000a
4
u
2
+ ··· + 1.39286a − 0.928571
−0.178571a
5
u
2
+ 0.250000a
4
u
2
+ ··· + 0.392857a − 0.928571
a
6
=
−0.0357143a
5
u
2
− 0.500000a
4
u
2
+ ··· − 1.32143a + 0.214286
1
14
a
5
u
2
−
1
4
a
4
u
2
+ ··· −
5
14
a +
4
7
a
10
=
3
14
a
5
u
2
+
1
4
a
4
u
2
+ ··· −
1
14
a −
9
7
0.535714a
5
u
2
+ 0.500000a
4
u
2
+ ··· + 0.0714286a − 1.71429
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
10
7
a
5
u
2
−
16
7
a
5
u + a
4
u
2
−
5
7
a
5
+ 3a
4
u −
3
7
a
3
u
2
+ a
4
+
33
7
a
3
u −
11
7
a
2
u
2
+
16
7
a
3
−
5
7
a
2
u −
5
7
u
2
a +
5
7
a
2
+
13
7
au −
6
7
u
2
−
13
7
a +
10
7
u +
32
7
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
18
− u
17
+ ··· − 16u − 8
c
2
, c
6
(u
3
+ u
2
+ 2u + 1)
6
c
3
, c
5
, c
9
c
10
u
18
− u
17
+ ··· + 64u − 8
c
7
, c
11
, c
12
(u
3
+ u
2
− 2u − 1)
6
c
8
(u
3
− 3u
2
− 4u − 1)
6
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
18
+ 19y
17
+ ··· − 3424y + 64
c
2
, c
6
(y
3
+ 3y
2
+ 2y −1)
6
c
3
, c
5
, c
9
c
10
y
18
− 5y
17
+ ··· − 2784y + 64
c
7
, c
11
, c
12
(y
3
− 5y
2
+ 6y −1)
6
c
8
(y
3
− 17y
2
+ 10y −1)
6
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.24698
a = −0.922052 + 0.388981I
b = −1.215350 − 0.381464I
−1.61418 + 2.82812I −1.50976 − 2.97945I
u = 1.24698
a = −0.922052 − 0.388981I
b = −1.215350 + 0.381464I
−1.61418 − 2.82812I −1.50976 + 2.97945I
u = 1.24698
a = 0.608898 + 0.654820I
b = −0.365738 + 0.960730I
−1.61418 − 2.82812I −1.50976 + 2.97945I
u = 1.24698
a = 0.608898 − 0.654820I
b = −0.365738 − 0.960730I
−1.61418 + 2.82812I −1.50976 − 2.97945I
u = 1.24698
a = 0.445706
b = 0.852713
2.52340 5.01950
u = 1.24698
a = −0.176294
b = 0.507529
2.52340 5.01950
u = −0.445042
a = −0.405582
b = −1.37094
−3.11638 5.01950
u = −0.445042
a = 0.43074 + 1.96960I
b = 1.62616 + 0.09419I
−7.25396 + 2.82812I −1.50976 − 2.97945I
u = −0.445042
a = 0.43074 − 1.96960I
b = 1.62616 − 0.09419I
−7.25396 − 2.82812I −1.50976 + 2.97945I
u = −0.445042
a = −2.65086
b = 0.429626
−3.11638 5.01950
15
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.445042
a = 3.12194 + 1.04628I
b = −0.532013 + 0.834636I
−7.25396 + 2.82812I −1.50976 − 2.97945I
u = −0.445042
a = 3.12194 − 1.04628I
b = −0.532013 − 0.834636I
−7.25396 − 2.82812I −1.50976 + 2.97945I
u = −1.80194
a = −0.411551 + 0.882060I
b = −0.43781 + 2.39534I
9.66536 + 2.82812I −1.50976 − 2.97945I
u = −1.80194
a = −0.411551 − 0.882060I
b = −0.43781 − 2.39534I
9.66536 − 2.82812I −1.50976 + 2.97945I
u = −1.80194
a = 0.261196 + 1.163730I
b = 0.16798 + 2.61487I
13.8029 5.01951 + 0.I
u = −1.80194
a = 0.261196 − 1.163730I
b = 0.16798 − 2.61487I
13.8029 5.01951 + 0.I
u = −1.80194
a = −0.195655 + 1.397520I
b = 0.04731 + 2.72683I
9.66536 − 2.82812I −1.50976 + 2.97945I
u = −1.80194
a = −0.195655 − 1.397520I
b = 0.04731 − 2.72683I
9.66536 + 2.82812I −1.50976 − 2.97945I
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
13
+ u
12
+ ··· + 3u
2
− 1)(u
18
− u
17
+ ··· − 16u − 8)
· (u
19
+ u
18
+ ··· + 12u + 1)
c
2
((u
3
+ u
2
+ 2u + 1)
6
)(u
13
− u
12
+ ··· + 2u + 1)
· (u
19
− 8u
18
+ ··· − 52u + 8)
c
3
, c
9
(u
13
+ u
12
− 4u
11
− 4u
10
+ 6u
9
+ 5u
8
− 2u
7
+ 4u
6
− 2u
5
− 14u
4
+ 7u
2
+ 1)
· (u
18
− u
17
+ ··· + 64u − 8)(u
19
− u
18
+ ··· − 2u − 1)
c
5
, c
10
(u
13
− u
12
− 4u
11
+ 4u
10
+ 6u
9
− 5u
8
− 2u
7
− 4u
6
− 2u
5
+ 14u
4
− 7u
2
− 1)
· (u
18
− u
17
+ ··· + 64u − 8)(u
19
− u
18
+ ··· − 2u − 1)
c
6
((u
3
+ u
2
+ 2u + 1)
6
)(u
13
+ u
12
+ ··· + 2u − 1)
· (u
19
− 8u
18
+ ··· − 52u + 8)
c
7
((u
3
+ u
2
− 2u − 1)
6
)(u
13
− 2u
12
+ ··· − 5u + 1)
· (u
19
− 9u
18
+ ··· + 12u − 8)
c
8
((u
3
− 3u
2
− 4u − 1)
6
)(u
13
+ 6u
12
+ ··· + u + 1)
· (u
19
+ 27u
18
+ ··· + 27116u + 3512)
c
11
, c
12
((u
3
+ u
2
− 2u − 1)
6
)(u
13
+ 2u
12
+ ··· − 5u − 1)
· (u
19
− 9u
18
+ ··· + 12u − 8)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
13
+ 5y
12
+ ··· + 6y −1)(y
18
+ 19y
17
+ ··· − 3424y + 64)
· (y
19
+ 35y
18
+ ··· + 42y −1)
c
2
, c
6
((y
3
+ 3y
2
+ 2y −1)
6
)(y
13
+ 13y
12
+ ··· + 18y −1)
· (y
19
+ 16y
18
+ ··· − 48y −64)
c
3
, c
5
, c
9
c
10
(y
13
− 9y
12
+ ··· − 14y −1)(y
18
− 5y
17
+ ··· − 2784y + 64)
· (y
19
− 11y
18
+ ··· + 10y −1)
c
7
, c
11
, c
12
((y
3
− 5y
2
+ 6y −1)
6
)(y
13
− 18y
12
+ ··· + 27y −1)
· (y
19
− 23y
18
+ ··· − 432y −64)
c
8
((y
3
− 17y
2
+ 10y −1)
6
)(y
13
− 26y
12
+ ··· + 43y −1)
· (y
19
− 43y
18
+ ··· + 4640976y −12334144)
18
