12n
0240
(K12n
0240
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 10 11 12 3 1 8 7 6
Solving Sequence
6,11
7 12
3,8
4 1 10 5 2 9
c
6
c
11
c
7
c
3
c
12
c
10
c
5
c
2
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
45
+ u
44
+ ··· + b + 2u, −2u
45
+ 2u
44
+ ··· + a + 8u, u
46
− 2u
45
+ ··· + u + 1i
I
u
2
= hu
3
+ b − u + 1, u
6
− 3u
4
+ 2u
2
+ a + 1, u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 54 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−u
45
+u
44
+· · ·+b+2u, −2u
45
+2u
44
+· · ·+a+8u, u
46
−2u
45
+· · ·+u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
12
=
−u
−u
3
+ u
a
3
=
2u
45
− 2u
44
+ ··· + 10u
2
− 8u
u
45
− u
44
+ ··· + 5u
2
− 2u
a
8
=
−u
2
+ 1
−u
4
+ 2u
2
a
4
=
4u
45
− 4u
44
+ ··· − 10u − 1
u
45
− u
44
+ ··· + 4u
2
− 3u
a
1
=
u
3
− 2u
−u
3
+ u
a
10
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
5
=
−u
12
+ 5u
10
− 9u
8
+ 6u
6
− u
2
+ 1
−u
14
+ 6u
12
− 13u
10
+ 10u
8
+ 2u
6
− 4u
4
− u
2
a
2
=
u
45
− u
44
+ ··· − 8u + 1
u
45
− u
44
+ ··· + 5u
2
− u
a
9
=
u
13
− 6u
11
+ 13u
9
− 10u
7
− 2u
5
+ 4u
3
+ u
−u
13
+ 5u
11
− 9u
9
+ 6u
7
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −3u
45
+ 56u
43
+ ··· − 5u − 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
46
+ 11u
45
+ ··· + 25u + 1
c
2
, c
4
u
46
− 9u
45
+ ··· − 9u + 1
c
3
, c
8
u
46
+ u
45
+ ··· + 1152u + 256
c
5
u
46
− 2u
45
+ ··· + 2660u + 1960
c
6
, c
7
, c
11
u
46
+ 2u
45
+ ··· − u + 1
c
9
u
46
+ 2u
45
+ ··· + 7u + 1
c
10
, c
12
u
46
− 6u
45
+ ··· − 73u + 17
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
46
+ 57y
45
+ ··· − 21y + 1
c
2
, c
4
y
46
− 11y
45
+ ··· − 25y + 1
c
3
, c
8
y
46
− 51y
45
+ ··· − 1490944y + 65536
c
5
y
46
+ 18y
45
+ ··· − 21367920y + 3841600
c
6
, c
7
, c
11
y
46
− 38y
45
+ ··· − 17y + 1
c
9
y
46
+ 54y
45
+ ··· − 17y + 1
c
10
, c
12
y
46
+ 34y
45
+ ··· − 5737y + 289
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.088878 + 0.844041I
a = 3.37742 − 1.22551I
b = −2.98228 + 0.61868I
11.39320 + 1.80249I −2.43037 − 0.91952I
u = −0.088878 − 0.844041I
a = 3.37742 + 1.22551I
b = −2.98228 − 0.61868I
11.39320 − 1.80249I −2.43037 + 0.91952I
u = −0.116026 + 0.837306I
a = −3.46305 + 1.35399I
b = 3.14660 − 0.77902I
10.46260 + 9.14915I −3.66280 − 5.53840I
u = −0.116026 − 0.837306I
a = −3.46305 − 1.35399I
b = 3.14660 + 0.77902I
10.46260 − 9.14915I −3.66280 + 5.53840I
u = 1.141830 + 0.219995I
a = −0.660574 + 0.098342I
b = −0.127713 − 0.217138I
−1.38478 − 0.54129I −5.73844 + 0.I
u = 1.141830 − 0.219995I
a = −0.660574 − 0.098342I
b = −0.127713 + 0.217138I
−1.38478 + 0.54129I −5.73844 + 0.I
u = 0.058004 + 0.794656I
a = 1.48803 − 0.66681I
b = −1.176270 + 0.234722I
3.97447 − 2.81253I −2.78364 + 4.01547I
u = 0.058004 − 0.794656I
a = 1.48803 + 0.66681I
b = −1.176270 − 0.234722I
3.97447 + 2.81253I −2.78364 − 4.01547I
u = −1.139460 + 0.393221I
a = −1.60826 + 1.63861I
b = 2.48711 + 1.35446I
7.33193 − 4.71190I −6.52326 + 0.I
u = −1.139460 − 0.393221I
a = −1.60826 − 1.63861I
b = 2.48711 − 1.35446I
7.33193 + 4.71190I −6.52326 + 0.I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.175260 + 0.396885I
a = 1.58950 − 1.52874I
b = −2.60724 − 1.21777I
8.06092 + 2.65955I 0
u = −1.175260 − 0.396885I
a = 1.58950 + 1.52874I
b = −2.60724 + 1.21777I
8.06092 − 2.65955I 0
u = −0.027447 + 0.752835I
a = −2.58406 − 0.48560I
b = 1.92495 + 0.96879I
1.16885 + 1.15848I −5.01072 + 0.12391I
u = −0.027447 − 0.752835I
a = −2.58406 + 0.48560I
b = 1.92495 − 0.96879I
1.16885 − 1.15848I −5.01072 − 0.12391I
u = 0.135112 + 0.737338I
a = −0.263549 − 0.833035I
b = 0.030415 + 0.576766I
1.53760 − 3.03900I −1.98360 + 5.04098I
u = 0.135112 − 0.737338I
a = −0.263549 + 0.833035I
b = 0.030415 − 0.576766I
1.53760 + 3.03900I −1.98360 − 5.04098I
u = 1.215480 + 0.337571I
a = 0.103389 + 1.085520I
b = −0.822594 + 0.083089I
0.425497 − 1.277230I 0
u = 1.215480 − 0.337571I
a = 0.103389 − 1.085520I
b = −0.822594 − 0.083089I
0.425497 + 1.277230I 0
u = −1.259210 + 0.309211I
a = −0.456708 + 1.219730I
b = 2.37986 − 0.29804I
−2.63767 + 2.66589I 0
u = −1.259210 − 0.309211I
a = −0.456708 − 1.219730I
b = 2.37986 + 0.29804I
−2.63767 − 2.66589I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.308600 + 0.052486I
a = 0.039504 + 0.254143I
b = 0.45316 − 1.39921I
−5.32887 + 1.91338I 0
u = −1.308600 − 0.052486I
a = 0.039504 − 0.254143I
b = 0.45316 + 1.39921I
−5.32887 − 1.91338I 0
u = 1.31477
a = 1.61451
b = 0.0640641
−6.82836 −12.1040
u = −0.494895 + 0.460674I
a = 0.357271 − 1.172810I
b = 0.450500 − 0.532533I
5.38865 + 5.18412I −6.71823 − 6.02395I
u = −0.494895 − 0.460674I
a = 0.357271 + 1.172810I
b = 0.450500 + 0.532533I
5.38865 − 5.18412I −6.71823 + 6.02395I
u = 1.291940 + 0.323151I
a = −1.15932 − 1.41965I
b = 1.49071 − 1.53239I
−2.95162 − 5.04688I 0
u = 1.291940 − 0.323151I
a = −1.15932 + 1.41965I
b = 1.49071 + 1.53239I
−2.95162 + 5.04688I 0
u = −0.418017 + 0.505684I
a = −0.149306 + 0.739200I
b = −0.653704 + 0.563087I
5.63155 − 1.66298I −5.86083 − 1.14993I
u = −0.418017 − 0.505684I
a = −0.149306 − 0.739200I
b = −0.653704 − 0.563087I
5.63155 + 1.66298I −5.86083 + 1.14993I
u = −1.307310 + 0.347945I
a = 0.790380 − 0.475352I
b = −1.45484 − 0.59474I
−0.29502 + 6.93232I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.307310 − 0.347945I
a = 0.790380 + 0.475352I
b = −1.45484 + 0.59474I
−0.29502 − 6.93232I 0
u = 1.366120 + 0.148240I
a = −0.533420 − 0.484870I
b = −0.989531 + 0.198012I
0.048597 − 0.512586I 0
u = 1.366120 − 0.148240I
a = −0.533420 + 0.484870I
b = −0.989531 − 0.198012I
0.048597 + 0.512586I 0
u = −1.345150 + 0.312918I
a = 0.253414 + 0.362467I
b = 0.191334 − 0.810458I
−3.12507 + 6.85143I 0
u = −1.345150 − 0.312918I
a = 0.253414 − 0.362467I
b = 0.191334 + 0.810458I
−3.12507 − 6.85143I 0
u = 1.329650 + 0.375534I
a = 0.49137 + 2.28669I
b = −3.08691 − 0.08101I
6.94740 − 6.18607I 0
u = 1.329650 − 0.375534I
a = 0.49137 − 2.28669I
b = −3.08691 + 0.08101I
6.94740 + 6.18607I 0
u = −1.38696
a = −0.181617
b = 0.699740
−7.20797 0
u = 1.345460 + 0.367374I
a = −0.43223 − 2.36028I
b = 3.46843 + 0.20726I
5.8692 − 13.4842I 0
u = 1.345460 − 0.367374I
a = −0.43223 + 2.36028I
b = 3.46843 − 0.20726I
5.8692 + 13.4842I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.390620 + 0.107679I
a = 0.514855 + 0.841670I
b = 0.807363 − 0.612141I
−0.55483 − 6.96949I 0
u = 1.390620 − 0.107679I
a = 0.514855 − 0.841670I
b = 0.807363 + 0.612141I
−0.55483 + 6.96949I 0
u = 0.582853
a = −0.782593
b = 0.178233
−1.21098 −7.83090
u = 0.282232 + 0.263140I
a = −0.99274 − 1.11009I
b = 0.157868 + 0.383616I
−0.549820 − 0.931505I −8.30910 + 7.33237I
u = 0.282232 − 0.263140I
a = −0.99274 + 1.11009I
b = 0.157868 − 0.383616I
−0.549820 + 0.931505I −8.30910 − 7.33237I
u = −0.263046
a = 2.94586
b = 0.883552
−2.04174 0.290920
9
II.
I
u
2
= hu
3
+b −u +1, u
6
−3u
4
+2u
2
+a +1, u
8
+u
7
−3u
6
−2u
5
+3u
4
+2u −1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
12
=
−u
−u
3
+ u
a
3
=
−u
6
+ 3u
4
− 2u
2
− 1
−u
3
+ u − 1
a
8
=
−u
2
+ 1
−u
4
+ 2u
2
a
4
=
−u
6
+ 3u
4
− 2u
2
− 1
−u
3
+ u − 1
a
1
=
u
3
− 2u
−u
3
+ u
a
10
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
5
=
−u
3
+ 2u
u
3
− u
a
2
=
−u
6
+ 3u
4
+ u
3
− 2u
2
− 2u − 1
−2u
3
+ 2u − 1
a
9
=
−u
2
+ 1
−u
4
+ 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
7
− 6u
6
+ 2u
5
+ 16u
4
− 5u
3
− 9u
2
+ 8u − 21
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
8
u
8
c
4
(u + 1)
8
c
5
, c
9
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
6
, c
7
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
10
, c
12
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
11
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
8
y
8
c
5
, c
9
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
6
, c
7
, c
11
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
10
, c
12
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.180120 + 0.268597I
a = −0.325934 + 0.693334I
b = −1.20799 − 0.83423I
−2.68559 − 1.13123I −10.92586 + 0.21647I
u = 1.180120 − 0.268597I
a = −0.325934 − 0.693334I
b = −1.20799 + 0.83423I
−2.68559 + 1.13123I −10.92586 − 0.21647I
u = 0.108090 + 0.747508I
a = 1.03462 − 0.99451I
b = −0.711982 + 1.138990I
0.51448 − 2.57849I −8.77377 + 3.25417I
u = 0.108090 − 0.747508I
a = 1.03462 + 0.99451I
b = −0.711982 − 1.138990I
0.51448 + 2.57849I −8.77377 − 3.25417I
u = −1.37100
a = −0.801005
b = 0.205997
−8.14766 −19.8990
u = −1.334530 + 0.318930I
a = 0.842429 − 0.289836I
b = −0.365014 − 1.352640I
−4.02461 + 6.44354I −14.3478 − 4.5473I
u = −1.334530 − 0.318930I
a = 0.842429 + 0.289836I
b = −0.365014 + 1.352640I
−4.02461 − 6.44354I −14.3478 + 4.5473I
u = 0.463640
a = −1.30123
b = −0.636025
−2.48997 −19.0060
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
46
+ 11u
45
+ ··· + 25u + 1)
c
2
((u − 1)
8
)(u
46
− 9u
45
+ ··· − 9u + 1)
c
3
, c
8
u
8
(u
46
+ u
45
+ ··· + 1152u + 256)
c
4
((u + 1)
8
)(u
46
− 9u
45
+ ··· − 9u + 1)
c
5
(u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1)
· (u
46
− 2u
45
+ ··· + 2660u + 1960)
c
6
, c
7
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
46
+ 2u
45
+ ··· − u + 1)
c
9
(u
8
− u
7
+ ··· + 2u − 1)(u
46
+ 2u
45
+ ··· + 7u + 1)
c
10
, c
12
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
46
− 6u
45
+ ··· − 73u + 17)
c
11
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
46
+ 2u
45
+ ··· − u + 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
46
+ 57y
45
+ ··· − 21y + 1)
c
2
, c
4
((y −1)
8
)(y
46
− 11y
45
+ ··· − 25y + 1)
c
3
, c
8
y
8
(y
46
− 51y
45
+ ··· − 1490944y + 65536)
c
5
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
46
+ 18y
45
+ ··· − 21367920y + 3841600)
c
6
, c
7
, c
11
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
46
− 38y
45
+ ··· − 17y + 1)
c
9
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
46
+ 54y
45
+ ··· − 17y + 1)
c
10
, c
12
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
46
+ 34y
45
+ ··· − 5737y + 289)
15
