12n
0676
(K12n
0676
)
A knot diagram
1
Linearized knot diagam
4 5 7 2 9 3 12 11 5 7 8 10
Solving Sequence
3,5
2 4
1,10
9 6 7 11 8 12
c
2
c
4
c
1
c
9
c
5
c
6
c
10
c
8
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−16813289u
16
− 180512824u
15
+ ··· + 375935808b + 173269993,
296972219u
16
+ 3308940840u
15
+ ··· + 1127807424a − 11385348987,
u
17
+ 11u
16
+ ··· − 24u + 1i
I
u
2
= ha
2
+ b + a + 2, a
3
+ 2a − 1, u − 1i
I
u
3
= hb
3
+ b
2
u + b
2
− 2u − 3, a, u
2
+ u − 1i
I
u
4
= h−a
3
− a
2
+ b − a − 2, a
4
+ a
3
+ 2a
2
+ 2a + 1, u − 1i
* 4 irreducible components of dim
C
= 0, with total 30 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−1.68 × 10
7
u
16
− 1.81 × 10
8
u
15
+ · · · + 3.76 × 10
8
b + 1.73 × 10
8
, 2.97 ×
10
8
u
16
+3.31×10
9
u
15
+· · ·+1.13×10
9
a−1.14×10
10
, u
17
+11u
16
+· · ·−24u+1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
4
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
10
=
−0.263318u
16
− 2.93396u
15
+ ··· + 47.5834u + 10.0951
0.0447238u
16
+ 0.480169u
15
+ ··· − 1.76545u − 0.460903
a
9
=
−0.263318u
16
− 2.93396u
15
+ ··· + 47.5834u + 10.0951
0.0407732u
16
+ 0.444464u
15
+ ··· − 2.40115u − 0.423444
a
6
=
−0.192384u
16
− 2.10577u
15
+ ··· + 12.3045u + 4.41593
−0.00859093u
16
− 0.0907645u
15
+ ··· − 0.527045u − 0.177073
a
7
=
−0.183793u
16
− 2.01500u
15
+ ··· + 12.8316u + 4.59300
−0.00859093u
16
− 0.0907645u
15
+ ··· − 0.527045u − 0.177073
a
11
=
−0.115128u
16
− 1.29525u
15
+ ··· + 24.3045u + 7.93474
0.0334181u
16
+ 0.371617u
15
+ ··· − 3.22037u − 0.304091
a
8
=
−0.157121u
16
− 1.73385u
15
+ ··· + 10.9663u + 7.16044
0.0206240u
16
+ 0.231250u
15
+ ··· − 4.03477u − 0.167498
a
12
=
0.183793u
16
+ 2.01500u
15
+ ··· − 12.8316u − 4.59300
−0.00282593u
16
− 0.0405560u
15
+ ··· + 3.13714u + 0.174058
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
69994879
187967904
u
16
−
2321600843
563903712
u
15
+ ··· +
17938178839
563903712
u +
1576601201
140975928
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
17
− 11u
16
+ ··· − 24u − 1
c
3
, c
6
u
17
+ 4u
16
+ ··· − 832u + 128
c
5
, c
9
u
17
− 2u
16
+ ··· + 224u + 64
c
7
, c
8
, c
11
u
17
+ 4u
16
+ ··· − 6u + 1
c
10
u
17
− 4u
16
+ ··· − 228u + 36
c
12
u
17
+ 10u
16
+ ··· + 4024u − 209
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
17
− 15y
16
+ ··· + 798y −1
c
3
, c
6
y
17
+ 36y
16
+ ··· + 487424y −16384
c
5
, c
9
y
17
− 28y
16
+ ··· + 50176y −4096
c
7
, c
8
, c
11
y
17
+ 14y
16
+ ··· + 46y −1
c
10
y
17
− 26y
16
+ ··· + 52488y −1296
c
12
y
17
− 42y
16
+ ··· + 23042342y −43681
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.770740 + 0.671347I
a = −0.997689 + 0.128313I
b = 0.36849 + 1.84869I
−1.47635 + 0.05755I −2.53546 − 1.02432I
u = 0.770740 − 0.671347I
a = −0.997689 − 0.128313I
b = 0.36849 − 1.84869I
−1.47635 − 0.05755I −2.53546 + 1.02432I
u = 1.020510 + 0.262233I
a = 0.189037 + 0.622417I
b = 0.31322 + 1.67490I
−4.25199 + 2.42274I −9.52858 − 5.40161I
u = 1.020510 − 0.262233I
a = 0.189037 − 0.622417I
b = 0.31322 − 1.67490I
−4.25199 − 2.42274I −9.52858 + 5.40161I
u = 0.820811
a = −0.456752
b = 1.11676
−1.14452 −10.7900
u = −1.51389 + 0.34866I
a = 0.676735 − 0.602259I
b = −0.66800 + 1.79459I
−11.88260 − 4.06700I −5.53941 + 2.68623I
u = −1.51389 − 0.34866I
a = 0.676735 + 0.602259I
b = −0.66800 − 1.79459I
−11.88260 + 4.06700I −5.53941 − 2.68623I
u = −0.125776 + 0.182671I
a = 0.73842 + 3.45151I
b = 0.252663 + 0.770640I
−3.46734 + 2.21457I 2.76869 − 2.41313I
u = −0.125776 − 0.182671I
a = 0.73842 − 3.45151I
b = 0.252663 − 0.770640I
−3.46734 − 2.21457I 2.76869 + 2.41313I
u = −1.87471
a = −0.672225
b = 2.12731
−7.11600 2.44290
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.0351827
a = 11.8728
b = −0.547681
0.826037 12.4770
u = −1.73829 + 1.01976I
a = −0.446194 − 1.097630I
b = 5.09465 − 1.32883I
7.23703 + 11.70460I −2.64244 − 4.71416I
u = −1.73829 − 1.01976I
a = −0.446194 + 1.097630I
b = 5.09465 + 1.32883I
7.23703 − 11.70460I −2.64244 + 4.71416I
u = −1.85948 + 1.35841I
a = 0.410440 + 1.181970I
b = −6.80560 + 2.10743I
12.6875 + 6.3547I 0.47634 − 2.59876I
u = −1.85948 − 1.35841I
a = 0.410440 − 1.181970I
b = −6.80560 − 2.10743I
12.6875 − 6.3547I 0.47634 + 2.59876I
u = −1.54446 + 1.96036I
a = −0.442677 − 1.276270I
b = 7.09639 − 6.41609I
9.80581 + 0.01769I −1.064012 + 0.845538I
u = −1.54446 − 1.96036I
a = −0.442677 + 1.276270I
b = 7.09639 + 6.41609I
9.80581 − 0.01769I −1.064012 − 0.845538I
6
II. I
u
2
= ha
2
+ b + a + 2, a
3
+ 2a − 1, u − 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
1
a
2
=
1
−1
a
4
=
1
0
a
1
=
0
−1
a
10
=
a
−a
2
− a − 2
a
9
=
a
−a
2
− 2
a
6
=
a
2
0
a
7
=
a
2
0
a
11
=
−a
2
− a + 1
−a
2
− a − 2
a
8
=
a
2
− 2a + 1
−1
a
12
=
a
2
−a
2
− 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 11a
2
+ 9a + 22
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
6
u
3
c
4
(u + 1)
3
c
5
, c
7
, c
8
u
3
+ 2u + 1
c
9
, c
11
, c
12
u
3
+ 2u − 1
c
10
u
3
− 3u
2
+ 5u − 2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
6
y
3
c
5
, c
7
, c
8
c
9
, c
11
, c
12
y
3
+ 4y
2
+ 4y −1
c
10
y
3
+ y
2
+ 13y −4
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.22670 + 1.46771I
b = 0.329484 − 0.802255I
−11.08570 − 5.13794I −3.17092 + 5.88938I
u = 1.00000
a = −0.22670 − 1.46771I
b = 0.329484 + 0.802255I
−11.08570 + 5.13794I −3.17092 − 5.88938I
u = 1.00000
a = 0.453398
b = −2.65897
−0.857735 28.3420
10
III. I
u
3
= hb
3
+ b
2
u + b
2
− 2u − 3, a, u
2
+ u − 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u − 1
a
4
=
u
−u + 1
a
1
=
u
−u
a
10
=
0
b
a
9
=
0
b
a
6
=
0
u
a
7
=
−u
u
a
11
=
−bu + b
bu
a
8
=
2b
2
u − b
2
− u
−b
2
u + b
2
+ b − 1
a
12
=
u
b
2
u − u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −b
2
u − 2b
2
− bu + 3b + 3u − 4
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
(u
2
+ u − 1)
3
c
4
, c
6
(u
2
− u − 1)
3
c
5
, c
9
u
6
c
7
, c
8
(u
3
+ u
2
+ 2u + 1)
2
c
10
, c
12
(u
3
+ u
2
− 1)
2
c
11
(u
3
− u
2
+ 2u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
(y
2
− 3y + 1)
3
c
5
, c
9
y
6
c
7
, c
8
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
10
, c
12
(y
3
− y
2
+ 2y −1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 0
b = 1.22142
0.126494 −3.14230
u = 0.618034
a = 0
b = −1.41973 + 1.20521I
−4.01109 − 2.82812I −7.00182 + 11.83005I
u = 0.618034
a = 0
b = −1.41973 − 1.20521I
−4.01109 + 2.82812I −7.00182 − 11.83005I
u = −1.61803
a = 0
b = 0.542287 + 0.460350I
−11.90680 + 2.82812I −6.38118 + 1.93520I
u = −1.61803
a = 0
b = 0.542287 − 0.460350I
−11.90680 − 2.82812I −6.38118 − 1.93520I
u = −1.61803
a = 0
b = −0.466540
−7.76919 −11.0920
14
IV. I
u
4
= h−a
3
− a
2
+ b − a − 2, a
4
+ a
3
+ 2a
2
+ 2a + 1, u − 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
1
a
2
=
1
−1
a
4
=
1
0
a
1
=
0
−1
a
10
=
a
a
3
+ a
2
+ a + 2
a
9
=
a
a
3
+ a
2
+ 2a + 2
a
6
=
a
2
0
a
7
=
a
2
0
a
11
=
−1
a
3
+ a
2
+ a + 2
a
8
=
−a
3
− a − 1
3a
3
+ a
2
+ 5a + 3
a
12
=
a
2
−a
2
− 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4a
3
− 3a
2
+ 4a − 4
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
4
c
3
, c
6
u
4
c
4
(u + 1)
4
c
5
, c
7
, c
8
u
4
− u
3
+ 2u
2
− 2u + 1
c
9
, c
11
, c
12
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
10
(u
2
+ u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
6
y
4
c
5
, c
7
, c
8
c
9
, c
11
, c
12
y
4
+ 3y
3
+ 2y
2
+ 1
c
10
(y
2
+ y + 1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.621744 + 0.440597I
b = 1.69244 + 0.31815I
−4.93480 − 2.02988I −6.57732 + 5.10773I
u = 1.00000
a = −0.621744 − 0.440597I
b = 1.69244 − 0.31815I
−4.93480 + 2.02988I −6.57732 − 5.10773I
u = 1.00000
a = 0.121744 + 1.306620I
b = −0.192440 − 0.547877I
−4.93480 + 2.02988I −0.92268 − 4.41855I
u = 1.00000
a = 0.121744 − 1.306620I
b = −0.192440 + 0.547877I
−4.93480 − 2.02988I −0.92268 + 4.41855I
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
7
)(u
2
+ u − 1)
3
(u
17
− 11u
16
+ ··· − 24u − 1)
c
3
u
7
(u
2
+ u − 1)
3
(u
17
+ 4u
16
+ ··· − 832u + 128)
c
4
((u + 1)
7
)(u
2
− u − 1)
3
(u
17
− 11u
16
+ ··· − 24u − 1)
c
5
u
6
(u
3
+ 2u + 1)(u
4
− u
3
+ ··· − 2u + 1)(u
17
− 2u
16
+ ··· + 224u + 64)
c
6
u
7
(u
2
− u − 1)
3
(u
17
+ 4u
16
+ ··· − 832u + 128)
c
7
, c
8
(u
3
+ 2u + 1)(u
3
+ u
2
+ 2u + 1)
2
(u
4
− u
3
+ 2u
2
− 2u + 1)
· (u
17
+ 4u
16
+ ··· − 6u + 1)
c
9
u
6
(u
3
+ 2u − 1)(u
4
+ u
3
+ ··· + 2u + 1)(u
17
− 2u
16
+ ··· + 224u + 64)
c
10
(u
2
+ u + 1)
2
(u
3
− 3u
2
+ 5u − 2)(u
3
+ u
2
− 1)
2
· (u
17
− 4u
16
+ ··· − 228u + 36)
c
11
(u
3
+ 2u − 1)(u
3
− u
2
+ 2u − 1)
2
(u
4
+ u
3
+ 2u
2
+ 2u + 1)
· (u
17
+ 4u
16
+ ··· − 6u + 1)
c
12
(u
3
+ 2u − 1)(u
3
+ u
2
− 1)
2
(u
4
+ u
3
+ 2u
2
+ 2u + 1)
· (u
17
+ 10u
16
+ ··· + 4024u − 209)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
7
)(y
2
− 3y + 1)
3
(y
17
− 15y
16
+ ··· + 798y −1)
c
3
, c
6
y
7
(y
2
− 3y + 1)
3
(y
17
+ 36y
16
+ ··· + 487424y −16384)
c
5
, c
9
y
6
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
17
− 28y
16
+ ··· + 50176y −4096)
c
7
, c
8
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
17
+ 14y
16
+ ··· + 46y −1)
c
10
(y
2
+ y + 1)
2
(y
3
− y
2
+ 2y −1)
2
(y
3
+ y
2
+ 13y −4)
· (y
17
− 26y
16
+ ··· + 52488y −1296)
c
12
(y
3
− y
2
+ 2y −1)
2
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
17
− 42y
16
+ ··· + 23042342y −43681)
20
