12n
0185
(K12n
0185
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 9 3 12 5 6 8 11
Solving Sequence
5,9
10 6 7
3,11
2 1 4 12 8
c
9
c
5
c
6
c
10
c
2
c
1
c
4
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h376024023995393u
34
+ 337812687444816u
33
+ ··· + 236220224312633b − 92299978153456,
− 21436667059562u
34
− 138041311777599u
33
+ ··· + 236220224312633a − 507133705138188,
u
35
+ 2u
34
+ ··· + 7u
2
− 1i
I
u
2
= h−u
7
− u
6
+ 2u
5
+ 3u
4
− 2u
2
+ b − 3u − 2, −u
7
+ u
6
+ 3u
5
− 2u
4
− 3u
3
+ a + 2,
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 43 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
= h3.76 × 10
14
u
34
+ 3.38 × 10
14
u
33
+ · · · + 2.36 × 10
14
b − 9.23 ×
10
13
, −2.14 × 10
13
u
34
− 1.38 × 10
14
u
33
+ · · · + 2.36 × 10
14
a − 5.07 ×
10
14
, u
35
+ 2u
34
+ · · · + 7u
2
− 1i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
u
2
a
6
=
−u
−u
3
+ u
a
7
=
u
3
− 2u
−u
3
+ u
a
3
=
0.0907487u
34
+ 0.584376u
33
+ ··· + 2.44395u + 2.14687
−1.59184u
34
− 1.43008u
33
+ ··· − 0.811363u + 0.390737
a
11
=
−u
2
+ 1
−u
4
+ 2u
2
a
2
=
0.0907487u
34
+ 0.584376u
33
+ ··· + 2.44395u + 2.14687
−1.45102u
34
− 1.18233u
33
+ ··· − 0.902112u − 0.0121412
a
1
=
0.575913u
34
+ 0.751783u
33
+ ··· + 1.72764u − 0.721893
0.273973u
34
+ 0.369123u
33
+ ··· + 0.836448u − 1.01092
a
4
=
−0.0507248u
34
+ 0.357239u
33
+ ··· + 1.91500u + 2.35136
−1.53287u
34
− 1.32266u
33
+ ··· − 0.588139u + 0.311133
a
12
=
0.356599u
34
+ 0.653291u
33
+ ··· + 1.26570u − 0.986457
0.215125u
34
+ 0.426155u
33
+ ··· + 0.736752u − 0.781963
a
8
=
−0.722473u
34
− 1.09863u
33
+ ··· − 1.64586u + 1.70851
0.119619u
34
+ 0.128063u
33
+ ··· − 0.342320u − 0.375739
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
2062473679663797
236220224312633
u
34
−
828575186416011
236220224312633
u
33
+ ··· +
1259833306856340
236220224312633
u −
231203744342586
236220224312633
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
35
+ 49u
34
+ ··· + 3150u + 1
c
2
, c
4
u
35
− 9u
34
+ ··· − 70u + 1
c
3
, c
7
u
35
− 3u
34
+ ··· − 2432u − 256
c
5
, c
9
, c
10
u
35
+ 2u
34
+ ··· + 7u
2
− 1
c
6
u
35
− 6u
34
+ ··· + 1364u − 847
c
8
, c
11
u
35
+ 2u
34
+ ··· + 4u + 1
c
12
u
35
+ 24u
34
+ ··· + 14u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
35
− 117y
34
+ ··· + 9175798y −1
c
2
, c
4
y
35
− 49y
34
+ ··· + 3150y −1
c
3
, c
7
y
35
+ 51y
34
+ ··· + 5423104y −65536
c
5
, c
9
, c
10
y
35
− 36y
34
+ ··· + 14y −1
c
6
y
35
− 36y
34
+ ··· + 23619926y −717409
c
8
, c
11
y
35
− 24y
34
+ ··· + 14y −1
c
12
y
35
− 24y
34
+ ··· + 402y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.571697 + 0.825819I
a = 1.02997 + 1.33115I
b = −0.13854 − 1.61473I
−8.58737 + 2.74263I −10.41024 − 2.56572I
u = −0.571697 − 0.825819I
a = 1.02997 − 1.33115I
b = −0.13854 + 1.61473I
−8.58737 − 2.74263I −10.41024 + 2.56572I
u = 0.552087 + 0.825280I
a = −0.87268 + 1.55881I
b = −0.20679 − 1.66620I
−12.94840 + 3.24222I −12.66532 − 0.54583I
u = 0.552087 − 0.825280I
a = −0.87268 − 1.55881I
b = −0.20679 + 1.66620I
−12.94840 − 3.24222I −12.66532 + 0.54583I
u = 0.575473 + 0.807100I
a = −1.29277 + 1.27295I
b = 0.43025 − 1.79381I
−13.0318 − 8.6571I −12.52203 + 5.43059I
u = 0.575473 − 0.807100I
a = −1.29277 − 1.27295I
b = 0.43025 + 1.79381I
−13.0318 + 8.6571I −12.52203 − 5.43059I
u = −1.215130 + 0.192273I
a = 0.340293 + 0.190807I
b = −0.450734 + 0.019324I
−1.72471 + 0.87161I −4.58345 + 0.49262I
u = −1.215130 − 0.192273I
a = 0.340293 − 0.190807I
b = −0.450734 − 0.019324I
−1.72471 − 0.87161I −4.58345 − 0.49262I
u = −0.128163 + 0.673108I
a = −0.157665 + 0.722315I
b = 0.179473 − 0.037885I
1.43459 + 2.27058I −1.42314 − 3.26969I
u = −0.128163 − 0.673108I
a = −0.157665 − 0.722315I
b = 0.179473 + 0.037885I
1.43459 − 2.27058I −1.42314 + 3.26969I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.521823 + 0.375425I
a = 1.98216 + 0.47145I
b = −0.419285 + 1.197250I
−4.17337 − 3.54764I −13.9483 + 7.1314I
u = 0.521823 − 0.375425I
a = 1.98216 − 0.47145I
b = −0.419285 − 1.197250I
−4.17337 + 3.54764I −13.9483 − 7.1314I
u = −0.628588
a = −2.61914
b = −0.676319
−6.22799 −17.7300
u = 1.38882
a = −0.0130861
b = 1.19281
−6.53287 −13.8260
u = 1.366140 + 0.266698I
a = −0.369885 + 0.416733I
b = 0.1060940 + 0.0070702I
−3.30368 − 5.68455I −8.00000 + 0.I
u = 1.366140 − 0.266698I
a = −0.369885 − 0.416733I
b = 0.1060940 − 0.0070702I
−3.30368 + 5.68455I −8.00000 + 0.I
u = −1.44080
a = −0.727727
b = 7.36351
−8.27744 27.9740
u = −1.47539
a = −0.857502
b = 2.36593
−8.21207 −8.00000
u = 1.47608 + 0.08600I
a = 0.570629 + 0.485997I
b = −0.33599 + 1.45625I
−6.62003 − 2.59519I 0
u = 1.47608 − 0.08600I
a = 0.570629 − 0.485997I
b = −0.33599 − 1.45625I
−6.62003 + 2.59519I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.349636 + 0.351362I
a = −1.224010 − 0.204907I
b = 0.210574 + 0.830362I
−0.594132 + 1.120540I −6.95383 − 6.15774I
u = −0.349636 − 0.351362I
a = −1.224010 + 0.204907I
b = 0.210574 − 0.830362I
−0.594132 − 1.120540I −6.95383 + 6.15774I
u = −1.52553 + 0.10391I
a = −0.768523 + 0.940824I
b = 0.459596 + 1.316590I
−10.99510 + 5.24946I 0
u = −1.52553 − 0.10391I
a = −0.768523 − 0.940824I
b = 0.459596 − 1.316590I
−10.99510 − 5.24946I 0
u = 1.54711
a = 1.47176
b = −0.141707
−13.5056 0
u = 0.209346 + 0.398609I
a = 0.21466 − 2.09699I
b = 0.49858 + 1.79535I
−3.31433 + 0.90397I −10.45251 + 5.11287I
u = 0.209346 − 0.398609I
a = 0.21466 + 2.09699I
b = 0.49858 − 1.79535I
−3.31433 − 0.90397I −10.45251 − 5.11287I
u = −0.439838
a = 0.227233
b = −0.497327
−0.965385 −10.6440
u = −1.56383 + 0.27842I
a = 1.192970 − 0.084215I
b = −0.57843 − 2.00327I
19.4383 + 12.6641I 0
u = −1.56383 − 0.27842I
a = 1.192970 + 0.084215I
b = −0.57843 + 2.00327I
19.4383 − 12.6641I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.56397 + 0.29270I
a = 1.145030 + 0.192290I
b = −0.07466 − 1.64853I
19.6118 + 0.8962I 0
u = −1.56397 − 0.29270I
a = 1.145030 − 0.192290I
b = −0.07466 + 1.64853I
19.6118 − 0.8962I 0
u = 1.56922 + 0.28522I
a = −1.121370 + 0.038644I
b = 0.40086 − 1.74127I
−15.6045 − 6.8496I 0
u = 1.56922 − 0.28522I
a = −1.121370 − 0.038644I
b = 0.40086 + 1.74127I
−15.6045 + 6.8496I 0
u = 0.344297
a = 3.18086
b = −0.768868
−2.11337 0.398900
8
II. I
u
2
= h−u
7
− u
6
+ 2u
5
+ 3u
4
− 2u
2
+ b − 3u − 2, −u
7
+ u
6
+ 3u
5
− 2u
4
−
3u
3
+ a + 2, u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
u
2
a
6
=
−u
−u
3
+ u
a
7
=
u
3
− 2u
−u
3
+ u
a
3
=
u
7
− u
6
− 3u
5
+ 2u
4
+ 3u
3
− 2
u
7
+ u
6
− 2u
5
− 3u
4
+ 2u
2
+ 3u + 2
a
11
=
−u
2
+ 1
−u
4
+ 2u
2
a
2
=
u
7
− u
6
− 3u
5
+ 2u
4
+ 3u
3
− 2
u
7
+ u
6
− 2u
5
− 3u
4
+ 2u
2
+ 2u + 2
a
1
=
0
−u
a
4
=
u
7
− u
6
− 3u
5
+ 2u
4
+ 3u
3
− 2
u
7
+ u
6
− 2u
5
− 3u
4
+ 2u
2
+ 3u + 2
a
12
=
−u
5
+ 2u
3
− u
−u
7
+ 3u
5
− 2u
3
− u
a
8
=
u
3
− 2u
−u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
7
− 9u
6
+ 10u
5
+ 27u
4
+ 2u
3
− 18u
2
− 20u − 29
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
7
u
8
c
4
(u + 1)
8
c
5
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
6
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
8
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
9
, c
10
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
11
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
12
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
8
c
3
, c
7
y
8
c
5
, c
9
, c
10
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
6
, c
12
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
c
8
, c
11
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.180120 + 0.268597I
a = −0.805639 − 0.183365I
b = 0.037144 + 0.630517I
−2.68559 + 1.13123I −13.38132 − 1.25921I
u = −1.180120 − 0.268597I
a = −0.805639 + 0.183365I
b = 0.037144 − 0.630517I
−2.68559 − 1.13123I −13.38132 + 1.25921I
u = −0.108090 + 0.747508I
a = −0.189481 − 1.310380I
b = 0.082879 + 0.802680I
0.51448 + 2.57849I −10.25723 − 4.63100I
u = −0.108090 − 0.747508I
a = −0.189481 + 1.310380I
b = 0.082879 − 0.802680I
0.51448 − 2.57849I −10.25723 + 4.63100I
u = 1.37100
a = 0.729394
b = 5.33104
−8.14766 −37.4550
u = 1.334530 + 0.318930I
a = 0.708845 − 0.169402I
b = −0.259819 + 0.832925I
−4.02461 − 6.44354I −13.7170 + 7.8762I
u = 1.334530 − 0.318930I
a = 0.708845 + 0.169402I
b = −0.259819 − 0.832925I
−4.02461 + 6.44354I −13.7170 − 7.8762I
u = −0.463640
a = −2.15684
b = 0.948553
−2.48997 −22.8330
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
35
+ 49u
34
+ ··· + 3150u + 1)
c
2
((u − 1)
8
)(u
35
− 9u
34
+ ··· − 70u + 1)
c
3
, c
7
u
8
(u
35
− 3u
34
+ ··· − 2432u − 256)
c
4
((u + 1)
8
)(u
35
− 9u
34
+ ··· − 70u + 1)
c
5
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
35
+ 2u
34
+ ··· + 7u
2
− 1)
c
6
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
35
− 6u
34
+ ··· + 1364u − 847)
c
8
(u
8
− u
7
+ ··· + 2u − 1)(u
35
+ 2u
34
+ ··· + 4u + 1)
c
9
, c
10
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
35
+ 2u
34
+ ··· + 7u
2
− 1)
c
11
(u
8
+ u
7
+ ··· − 2u − 1)(u
35
+ 2u
34
+ ··· + 4u + 1)
c
12
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
35
+ 24u
34
+ ··· + 14u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
8
)(y
35
− 117y
34
+ ··· + 9175798y − 1)
c
2
, c
4
((y − 1)
8
)(y
35
− 49y
34
+ ··· + 3150y − 1)
c
3
, c
7
y
8
(y
35
+ 51y
34
+ ··· + 5423104y − 65536)
c
5
, c
9
, c
10
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
35
− 36y
34
+ ··· + 14y − 1)
c
6
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
35
− 36y
34
+ ··· + 23619926y − 717409)
c
8
, c
11
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
35
− 24y
34
+ ··· + 14y − 1)
c
12
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
35
− 24y
34
+ ··· + 402y − 1)
14
