12n
0116
(K12n
0116
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 10 3 12 11 5 7 8 9
Solving Sequence
5,9
10
3,6
7 11 2 1 4 8 12
c
9
c
5
c
6
c
10
c
2
c
1
c
4
c
8
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.49041 × 10
48
u
40
− 1.63923 × 10
48
u
39
+ ··· + 4.94994 × 10
48
b + 1.35928 × 10
48
,
− 1.49041 × 10
48
u
40
− 1.63923 × 10
48
u
39
+ ··· + 4.94994 × 10
48
a + 1.35928 × 10
48
, u
41
+ 2u
40
+ ··· − u − 1i
I
u
2
= h−u
5
+ u
4
+ 3u
3
− 2u
2
+ b − u − 1, −u
5
+ u
4
+ 3u
3
− 2u
2
+ a − 2u − 1, u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 47 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−1.49×10
48
u
40
−1.64×10
48
u
39
+· · ·+4.95×10
48
b+1.36×10
48
, −1.49×
10
48
u
40
−1.64×10
48
u
39
+· · ·+4.95×10
48
a+1.36×10
48
, u
41
+2u
40
+· · ·−u−1i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
3
=
0.301096u
40
+ 0.331162u
39
+ ··· + 2.34511u − 0.274605
0.301096u
40
+ 0.331162u
39
+ ··· + 3.34511u − 0.274605
a
6
=
u
−u
3
+ u
a
7
=
−0.501619u
40
− 0.639447u
39
+ ··· + 0.896473u + 0.455440
−0.458393u
40
− 0.639616u
39
+ ··· + 0.799469u + 0.637044
a
11
=
−0.205384u
40
− 0.351951u
39
+ ··· + 0.606975u + 1.47006
−0.559530u
40
− 0.970963u
39
+ ··· + 1.39792u + 1.11199
a
2
=
0.301096u
40
+ 0.331162u
39
+ ··· + 2.34511u − 0.274605
0.0224886u
40
− 0.0444496u
39
+ ··· + 3.31504u − 0.00357471
a
1
=
0.110835u
40
− 0.0229094u
39
+ ··· − 0.234831u + 0.545396
−0.390784u
40
− 0.662356u
39
+ ··· + 0.661642u + 1.00084
a
4
=
0.257870u
40
+ 0.331331u
39
+ ··· + 2.44211u − 0.456209
0.313467u
40
+ 0.443108u
39
+ ··· + 3.39872u − 0.542830
a
8
=
−0.659770u
40
− 1.09530u
39
+ ··· + 1.32120u + 1.69134
−0.119568u
40
+ 0.0303721u
39
+ ··· − 0.186514u − 1.02623
a
12
=
0.501619u
40
+ 0.639447u
39
+ ··· − 0.896473u − 0.455440
−0.390784u
40
− 0.662356u
39
+ ··· + 0.661642u + 1.00084
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4.91398u
40
− 6.25021u
39
+ ··· − 14.4903u + 31.2535
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
41
+ 47u
40
+ ··· + 296u + 1
c
2
, c
4
u
41
− 7u
40
+ ··· − 12u + 1
c
3
, c
6
u
41
+ 7u
40
+ ··· + 640u − 64
c
5
, c
9
u
41
+ 2u
40
+ ··· − u − 1
c
7
, c
8
, c
11
u
41
+ 2u
40
+ ··· + u + 1
c
10
, c
12
u
41
− 2u
40
+ ··· − 47u + 17
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
41
− 99y
40
+ ··· + 111528y −1
c
2
, c
4
y
41
− 47y
40
+ ··· + 296y −1
c
3
, c
6
y
41
+ 39y
40
+ ··· + 90112y −4096
c
5
, c
9
y
41
+ 42y
39
+ ··· + 7y −1
c
7
, c
8
, c
11
y
41
+ 36y
40
+ ··· + 7y −1
c
10
, c
12
y
41
− 12y
40
+ ··· − 2041y −289
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.546127 + 0.811291I
a = 0.743645 − 1.202480I
b = 0.197518 − 0.391191I
−4.83591 − 7.62929I 1.63110 + 8.21040I
u = −0.546127 − 0.811291I
a = 0.743645 + 1.202480I
b = 0.197518 + 0.391191I
−4.83591 + 7.62929I 1.63110 − 8.21040I
u = 0.559771 + 0.753240I
a = −0.624352 − 1.085930I
b = −0.064581 − 0.332685I
0.18158 + 4.25829I 7.22708 − 8.04646I
u = 0.559771 − 0.753240I
a = −0.624352 + 1.085930I
b = −0.064581 + 0.332685I
0.18158 − 4.25829I 7.22708 + 8.04646I
u = −0.664122 + 0.603424I
a = 0.468587 − 0.663053I
b = −0.195535 − 0.059628I
−2.14123 − 1.43316I 6.02200 + 3.64151I
u = −0.664122 − 0.603424I
a = 0.468587 + 0.663053I
b = −0.195535 + 0.059628I
−2.14123 + 1.43316I 6.02200 − 3.64151I
u = 0.350975 + 0.790999I
a = −0.39756 − 1.62860I
b = −0.046582 − 0.837603I
−6.73300 + 0.22938I −2.25086 − 2.33731I
u = 0.350975 − 0.790999I
a = −0.39756 + 1.62860I
b = −0.046582 + 0.837603I
−6.73300 − 0.22938I −2.25086 + 2.33731I
u = 0.087719 + 0.850756I
a = −0.12108 − 2.10269I
b = −0.033358 − 1.251940I
−8.03114 + 3.86698I −3.56668 − 3.96784I
u = 0.087719 − 0.850756I
a = −0.12108 + 2.10269I
b = −0.033358 + 1.251940I
−8.03114 − 3.86698I −3.56668 + 3.96784I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.445566 + 0.616611I
a = 0.112439 − 1.137990I
b = −0.333128 − 0.521377I
−1.53709 − 1.40851I 1.34634 + 3.01002I
u = −0.445566 − 0.616611I
a = 0.112439 + 1.137990I
b = −0.333128 + 0.521377I
−1.53709 + 1.40851I 1.34634 − 3.01002I
u = −0.090722 + 0.752674I
a = 0.05605 − 1.99845I
b = −0.034673 − 1.245770I
−2.86548 − 1.28813I 1.16839 + 4.84793I
u = −0.090722 − 0.752674I
a = 0.05605 + 1.99845I
b = −0.034673 + 1.245770I
−2.86548 + 1.28813I 1.16839 − 4.84793I
u = −0.527293 + 0.436682I
a = −0.973519 − 0.176954I
b = −1.50081 + 0.25973I
−4.01492 + 3.79256I 2.77393 − 0.06472I
u = −0.527293 − 0.436682I
a = −0.973519 + 0.176954I
b = −1.50081 − 0.25973I
−4.01492 − 3.79256I 2.77393 + 0.06472I
u = 0.468495 + 0.464998I
a = 0.622134 − 0.271062I
b = 1.090630 + 0.193936I
0.599224 − 0.610593I 8.50615 − 0.01136I
u = 0.468495 − 0.464998I
a = 0.622134 + 0.271062I
b = 1.090630 − 0.193936I
0.599224 + 0.610593I 8.50615 + 0.01136I
u = −0.325958 + 0.548034I
a = −0.267538 + 0.084708I
b = −0.593496 + 0.632743I
−2.79420 − 2.18222I 4.51636 + 3.99594I
u = −0.325958 − 0.548034I
a = −0.267538 − 0.084708I
b = −0.593496 − 0.632743I
−2.79420 + 2.18222I 4.51636 − 3.99594I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.397250 + 0.182037I
a = −0.506275 − 0.004273I
b = 0.890974 + 0.177764I
2.57061 + 4.43167I 14.03660 + 0.I
u = 1.397250 − 0.182037I
a = −0.506275 + 0.004273I
b = 0.890974 − 0.177764I
2.57061 − 4.43167I 14.03660 + 0.I
u = −1.41231
a = 0.496036
b = −0.916272
6.54065 18.5250
u = 0.487776 + 0.155470I
a = 2.02925 − 0.39422I
b = 2.51703 − 0.23875I
−4.99926 + 2.47497I 10.9995 − 16.9397I
u = 0.487776 − 0.155470I
a = 2.02925 + 0.39422I
b = 2.51703 + 0.23875I
−4.99926 − 2.47497I 10.9995 + 16.9397I
u = 0.497461
a = −0.110298
b = 0.387162
0.683544 14.8150
u = −1.05090 + 1.14678I
a = 0.504687 + 0.968425I
b = −0.54621 + 2.11520I
−11.9065 − 13.7324I 0
u = −1.05090 − 1.14678I
a = 0.504687 − 0.968425I
b = −0.54621 − 2.11520I
−11.9065 + 13.7324I 0
u = 1.05595 + 1.16152I
a = −0.501764 + 0.917028I
b = 0.55418 + 2.07855I
−6.55369 + 9.57234I 0
u = 1.05595 − 1.16152I
a = −0.501764 − 0.917028I
b = 0.55418 − 2.07855I
−6.55369 − 9.57234I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.10127 + 1.14227I
a = −0.673483 + 0.861835I
b = 0.42779 + 2.00411I
−16.5464 + 4.1585I 0
u = 1.10127 − 1.14227I
a = −0.673483 − 0.861835I
b = 0.42779 − 2.00411I
−16.5464 − 4.1585I 0
u = −1.07763 + 1.17451I
a = 0.535143 + 0.849624I
b = −0.54248 + 2.02413I
−8.52808 − 4.96231I 0
u = −1.07763 − 1.17451I
a = 0.535143 − 0.849624I
b = −0.54248 − 2.02413I
−8.52808 + 4.96231I 0
u = −1.16301 + 1.12527I
a = 0.754065 + 0.641582I
b = −0.40895 + 1.76685I
−11.60780 + 5.44383I 0
u = −1.16301 − 1.12527I
a = 0.754065 − 0.641582I
b = −0.40895 − 1.76685I
−11.60780 − 5.44383I 0
u = −0.373875
a = −2.77728
b = −3.15116
−0.771990 64.9410
u = −1.13639 + 1.17348I
a = 0.623222 + 0.717077I
b = −0.51317 + 1.89056I
−8.37413 − 3.52292I 0
u = −1.13639 − 1.17348I
a = 0.623222 − 0.717077I
b = −0.51317 − 1.89056I
−8.37413 + 3.52292I 0
u = 1.16288 + 1.15013I
a = −0.687885 + 0.655673I
b = 0.47499 + 1.80580I
−6.27214 − 1.18033I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.16288 − 1.15013I
a = −0.687885 − 0.655673I
b = 0.47499 − 1.80580I
−6.27214 + 1.18033I 0
9
II. I
u
2
= h−u
5
+ u
4
+ 3u
3
− 2u
2
+ b − u − 1, −u
5
+ u
4
+ 3u
3
− 2u
2
+ a −
2u − 1, u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
3
=
u
5
− u
4
− 3u
3
+ 2u
2
+ 2u + 1
u
5
− u
4
− 3u
3
+ 2u
2
+ u + 1
a
6
=
u
−u
3
+ u
a
7
=
u
−u
3
+ u
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
u
5
− u
4
− 3u
3
+ 2u
2
+ 2u + 1
u
5
− u
4
− 3u
3
+ 2u
2
+ 1
a
1
=
0
−u
a
4
=
u
5
− u
4
− 3u
3
+ 2u
2
+ 2u + 1
u
5
− u
4
− 3u
3
+ 2u
2
+ u + 1
a
8
=
−u
5
+ 2u
3
+ u
u
5
− 3u
3
+ u
a
12
=
u
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −7u
5
+ 3u
4
+ 19u
3
− 5u
2
− 8u − 6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
6
u
6
c
4
(u + 1)
6
c
5
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
7
, c
8
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
9
, c
10
, c
12
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
11
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
6
y
6
c
5
, c
9
, c
10
c
12
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
7
, c
8
, c
11
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.493180 + 0.575288I
a = −0.858925 − 1.001920I
b = −0.36575 − 1.57721I
−4.60518 − 1.97241I 3.78159 + 4.50121I
u = −0.493180 − 0.575288I
a = −0.858925 + 1.001920I
b = −0.36575 + 1.57721I
−4.60518 + 1.97241I 3.78159 − 4.50121I
u = 0.483672
a = 2.06752
b = 1.58384
−0.906083 −8.91030
u = 1.52087 + 0.16310I
a = 0.650045 − 0.069710I
b = −0.870821 − 0.232805I
2.05064 + 4.59213I −0.56679 − 5.39767I
u = 1.52087 − 0.16310I
a = 0.650045 + 0.069710I
b = −0.870821 + 0.232805I
2.05064 − 4.59213I −0.56679 + 5.39767I
u = −1.53904
a = −0.649754
b = 0.889289
6.01515 2.48070
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
41
+ 47u
40
+ ··· + 296u + 1)
c
2
((u − 1)
6
)(u
41
− 7u
40
+ ··· − 12u + 1)
c
3
, c
6
u
6
(u
41
+ 7u
40
+ ··· + 640u − 64)
c
4
((u + 1)
6
)(u
41
− 7u
40
+ ··· − 12u + 1)
c
5
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
41
+ 2u
40
+ ··· − u − 1)
c
7
, c
8
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
41
+ 2u
40
+ ··· + u + 1)
c
9
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
41
+ 2u
40
+ ··· − u − 1)
c
10
, c
12
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
41
− 2u
40
+ ··· − 47u + 17)
c
11
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
41
+ 2u
40
+ ··· + u + 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
41
− 99y
40
+ ··· + 111528y −1)
c
2
, c
4
((y −1)
6
)(y
41
− 47y
40
+ ··· + 296y −1)
c
3
, c
6
y
6
(y
41
+ 39y
40
+ ··· + 90112y −4096)
c
5
, c
9
(y
6
− 7y
5
+ ··· − 5y + 1)(y
41
+ 42y
39
+ ··· + 7y −1)
c
7
, c
8
, c
11
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
41
+ 36y
40
+ ··· + 7y −1)
c
10
, c
12
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
41
− 12y
40
+ ··· − 2041y −289)
15
