12n
0108
(K12n
0108
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 9 3 12 11 5 7 8 10
Solving Sequence
3,5
2 1
4,10
9 6 7 11 8 12
c
2
c
1
c
4
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−9.27106 × 10
41
u
40
− 5.88994 × 10
42
u
39
+ ··· + 9.61206 × 10
40
b − 6.11993 × 10
41
,
− 6.34476 × 10
41
u
40
− 4.08468 × 10
42
u
39
+ ··· + 9.61206 × 10
40
a − 1.64646 × 10
42
,
u
41
+ 8u
40
+ ··· + 34u + 1i
I
u
2
= ha
2
+ b − a + 2, a
3
+ 2a + 1, u − 1i
I
u
3
= ha
3
− a
2
+ b + a − 2, a
4
− a
3
+ 2a
2
− 2a + 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 48 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−9.27 × 10
41
u
40
− 5.89 × 10
42
u
39
+ · · · + 9.61 × 10
40
b − 6.12 ×
10
41
, −6.34 × 10
41
u
40
− 4.08 × 10
42
u
39
+ · · · + 9.61 × 10
40
a − 1.65 ×
10
42
, u
41
+ 8u
40
+ · · · + 34u + 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
4
=
u
−u
3
+ u
a
10
=
6.60083u
40
+ 42.4953u
39
+ ··· + 119.918u + 17.1291
9.64523u
40
+ 61.2765u
39
+ ··· + 197.232u + 6.36693
a
9
=
6.60083u
40
+ 42.4953u
39
+ ··· + 119.918u + 17.1291
−7.48281u
40
− 47.2910u
39
+ ··· − 146.752u − 3.94440
a
6
=
−1.00706u
40
− 6.01141u
39
+ ··· − 9.30914u + 5.53712
1.60537u
40
+ 9.92359u
39
+ ··· + 28.7480u + 1.03801
a
7
=
0.598315u
40
+ 3.91218u
39
+ ··· + 19.4389u + 6.57513
1.60537u
40
+ 9.92359u
39
+ ··· + 28.7480u + 1.03801
a
11
=
4.48614u
40
+ 28.6835u
39
+ ··· + 68.1177u + 12.1782
−6.77071u
40
− 42.6663u
39
+ ··· − 132.736u − 3.61078
a
8
=
2.18696u
40
+ 13.8857u
39
+ ··· + 34.8186u + 9.92100
−0.720306u
40
− 4.14288u
39
+ ··· − 11.5196u − 0.0728726
a
12
=
−0.598315u
40
− 3.91218u
39
+ ··· − 19.4389u − 6.57513
1.49635u
40
+ 9.28064u
39
+ ··· + 32.3421u + 0.715717
(ii) Obstruction class = −1
(iii) Cusp Shapes = 8.64366u
40
+ 54.7705u
39
+ ··· + 150.271u − 7.40030
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
41
+ 50u
40
+ ··· + 1026u + 1
c
2
, c
4
u
41
− 8u
40
+ ··· + 34u − 1
c
3
, c
6
u
41
+ 7u
40
+ ··· + 448u + 128
c
5
, c
9
u
41
+ 2u
40
+ ··· − u − 1
c
7
, c
8
, c
11
u
41
− 2u
40
+ ··· − 3u − 1
c
10
u
41
+ 2u
40
+ ··· − 240u − 36
c
12
u
41
− 12u
40
+ ··· − 467u + 163
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
41
− 110y
40
+ ··· + 1177934y −1
c
2
, c
4
y
41
− 50y
40
+ ··· + 1026y −1
c
3
, c
6
y
41
+ 45y
40
+ ··· + 520192y −16384
c
5
, c
9
y
41
+ 42y
39
+ ··· + 11y −1
c
7
, c
8
, c
11
y
41
+ 36y
40
+ ··· + 11y −1
c
10
y
41
− 12y
40
+ ··· + 7992y −1296
c
12
y
41
− 12y
40
+ ··· + 2120951y −26569
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.023890 + 0.144827I
a = −0.114598 + 0.492587I
b = −0.25937 − 2.41362I
1.10253 + 2.40302I −1.4441 − 16.6541I
u = 1.023890 − 0.144827I
a = −0.114598 − 0.492587I
b = −0.25937 + 2.41362I
1.10253 − 2.40302I −1.4441 + 16.6541I
u = 1.09654
a = 0.432301
b = 1.83917
−2.21383 3.32100
u = 0.799614 + 0.163499I
a = 0.305351 − 0.647956I
b = 0.24502 + 1.73105I
−2.56053 − 0.52413I −15.2569 − 4.2496I
u = 0.799614 − 0.163499I
a = 0.305351 + 0.647956I
b = 0.24502 − 1.73105I
−2.56053 + 0.52413I −15.2569 + 4.2496I
u = 0.618990 + 0.472493I
a = −0.934348 − 0.002226I
b = −0.492958 + 0.445400I
−0.79846 − 1.42488I −6.70258 + 4.89942I
u = 0.618990 − 0.472493I
a = −0.934348 + 0.002226I
b = −0.492958 − 0.445400I
−0.79846 + 1.42488I −6.70258 − 4.89942I
u = 0.654305 + 0.324584I
a = −0.185932 + 0.902814I
b = 0.19586 − 1.59294I
1.89593 − 3.76450I −6.51510 − 0.37648I
u = 0.654305 − 0.324584I
a = −0.185932 − 0.902814I
b = 0.19586 + 1.59294I
1.89593 + 3.76450I −6.51510 + 0.37648I
u = 0.024088 + 0.729003I
a = 0.975104 − 0.793348I
b = 0.015225 − 0.189698I
4.22621 − 1.38545I −1.96542 + 3.48117I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.024088 − 0.729003I
a = 0.975104 + 0.793348I
b = 0.015225 + 0.189698I
4.22621 + 1.38545I −1.96542 − 3.48117I
u = −0.690147 + 0.145492I
a = 0.04771 − 1.93802I
b = −0.236345 − 0.828778I
8.34723 − 4.89832I 3.94388 + 1.15377I
u = −0.690147 − 0.145492I
a = 0.04771 + 1.93802I
b = −0.236345 + 0.828778I
8.34723 + 4.89832I 3.94388 − 1.15377I
u = 0.784586 + 1.118630I
a = 0.619859 − 0.276128I
b = 0.550920 + 0.122912I
−3.61696 − 3.86307I 0
u = 0.784586 − 1.118630I
a = 0.619859 + 0.276128I
b = 0.550920 − 0.122912I
−3.61696 + 3.86307I 0
u = 1.06503 + 0.93172I
a = −0.595278 + 0.164576I
b = −0.773274 − 0.052105I
−0.499621 − 0.427314I 0
u = 1.06503 − 0.93172I
a = −0.595278 − 0.164576I
b = −0.773274 + 0.052105I
−0.499621 + 0.427314I 0
u = 0.64097 + 1.26233I
a = −0.593984 + 0.346167I
b = −0.439012 − 0.204480I
0.95653 − 7.53305I 0
u = 0.64097 − 1.26233I
a = −0.593984 − 0.346167I
b = −0.439012 + 0.204480I
0.95653 + 7.53305I 0
u = −0.546543 + 0.110125I
a = −0.15010 + 2.08713I
b = 0.113534 + 0.680577I
2.42231 − 1.86356I −0.33827 + 3.07051I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.546543 − 0.110125I
a = −0.15010 − 2.08713I
b = 0.113534 − 0.680577I
2.42231 + 1.86356I −0.33827 − 3.07051I
u = −1.52433 + 0.18776I
a = −0.857605 + 0.620146I
b = −1.95651 + 0.55267I
−1.30698 + 4.28669I 0
u = −1.52433 − 0.18776I
a = −0.857605 − 0.620146I
b = −1.95651 − 0.55267I
−1.30698 − 4.28669I 0
u = −1.64128 + 0.09999I
a = 0.638107 + 0.741558I
b = 1.78817 + 0.37849I
−6.19266 + 5.42860I 0
u = −1.64128 − 0.09999I
a = 0.638107 − 0.741558I
b = 1.78817 − 0.37849I
−6.19266 − 5.42860I 0
u = −1.68045 + 0.02735I
a = −0.661327 − 0.687909I
b = −1.85642 − 0.38244I
−11.48150 + 1.16744I 0
u = −1.68045 − 0.02735I
a = −0.661327 + 0.687909I
b = −1.85642 + 0.38244I
−11.48150 − 1.16744I 0
u = −1.68637 + 0.08037I
a = 0.707793 − 0.630890I
b = 1.93021 − 0.41903I
−9.39900 + 3.48257I 0
u = −1.68637 − 0.08037I
a = 0.707793 + 0.630890I
b = 1.93021 + 0.41903I
−9.39900 − 3.48257I 0
u = −1.69925 + 0.27352I
a = 0.758442 − 0.516353I
b = 2.04847 − 0.47151I
−9.49392 + 4.85059I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.69925 − 0.27352I
a = 0.758442 + 0.516353I
b = 2.04847 + 0.47151I
−9.49392 − 4.85059I 0
u = −1.67351 + 0.43283I
a = 0.792521 − 0.425685I
b = 2.12131 − 0.52943I
−6.4546 + 13.6928I 0
u = −1.67351 − 0.43283I
a = 0.792521 + 0.425685I
b = 2.12131 + 0.52943I
−6.4546 − 13.6928I 0
u = −1.70062 + 0.36989I
a = −0.772140 + 0.460082I
b = −2.09871 + 0.49854I
−11.6676 + 9.4552I 0
u = −1.70062 − 0.36989I
a = −0.772140 − 0.460082I
b = −2.09871 − 0.49854I
−11.6676 − 9.4552I 0
u = 1.79083
a = −0.471299
b = −1.27778
−5.96639 0
u = 1.80924 + 0.21929I
a = 0.470409 − 0.020609I
b = 1.266140 + 0.027083I
−2.04881 − 3.64658I 0
u = 1.80924 − 0.21929I
a = 0.470409 + 0.020609I
b = 1.266140 − 0.027083I
−2.04881 + 3.64658I 0
u = −0.006712 + 0.160531I
a = −1.90793 − 3.46251I
b = −0.690720 + 0.504582I
3.36845 + 2.26324I −4.13576 − 3.78467I
u = −0.006712 − 0.160531I
a = −1.90793 + 3.46251I
b = −0.690720 − 0.504582I
3.36845 − 2.26324I −4.13576 + 3.78467I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.0303735
a = 13.9549
b = 0.495519
−0.822843 −12.1130
9
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
1
=
0
−1
a
4
=
1
0
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 − 34
10
(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
11
(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
12
(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
7.79580 − 5.13794I −8.82908 + 5.88938I
u = 1.00000
a = 0.22670 − 1.46771I
b = 0.329484 − 0.802255I
7.79580 + 5.13794I −8.82908 − 5.88938I
u = 1.00000
a = −0.453398
b = −2.65897
−2.43213 −40.3420
13
III. I
u
3
= ha
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
1
=
0
−1
a
4
=
1
0
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 − 8
14
(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
15
(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
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.621744 + 0.440597I
b = 1.69244 − 0.31815I
1.64493 − 2.02988I −5.42268 + 5.10773I
u = 1.00000
a = 0.621744 − 0.440597I
b = 1.69244 + 0.31815I
1.64493 + 2.02988I −5.42268 − 5.10773I
u = 1.00000
a = −0.121744 + 1.306620I
b = −0.192440 + 0.547877I
1.64493 + 2.02988I −11.07732 − 4.41855I
u = 1.00000
a = −0.121744 − 1.306620I
b = −0.192440 − 0.547877I
1.64493 − 2.02988I −11.07732 + 4.41855I
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
7
)(u
41
+ 50u
40
+ ··· + 1026u + 1)
c
2
((u − 1)
7
)(u
41
− 8u
40
+ ··· + 34u − 1)
c
3
, c
6
u
7
(u
41
+ 7u
40
+ ··· + 448u + 128)
c
4
((u + 1)
7
)(u
41
− 8u
40
+ ··· + 34u − 1)
c
5
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
41
+ 2u
40
+ ··· − u − 1)
c
7
, c
8
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
41
− 2u
40
+ ··· − 3u − 1)
c
9
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
41
+ 2u
40
+ ··· − u − 1)
c
10
((u
2
− u + 1)
2
)(u
3
+ 3u
2
+ 5u + 2)(u
41
+ 2u
40
+ ··· − 240u − 36)
c
11
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
41
− 2u
40
+ ··· − 3u − 1)
c
12
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
41
− 12u
40
+ ··· − 467u + 163)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
7
)(y
41
− 110y
40
+ ··· + 1177934y −1)
c
2
, c
4
((y −1)
7
)(y
41
− 50y
40
+ ··· + 1026y −1)
c
3
, c
6
y
7
(y
41
+ 45y
40
+ ··· + 520192y −16384)
c
5
, c
9
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
41
+ 42y
39
+ ··· + 11y −1)
c
7
, c
8
, c
11
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
41
+ 36y
40
+ ··· + 11y −1)
c
10
((y
2
+ y + 1)
2
)(y
3
+ y
2
+ 13y −4)(y
41
− 12y
40
+ ··· + 7992y −1296)
c
12
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
41
− 12y
40
+ ··· + 2120951y −26569)
19
