12n
0688
(K12n
0688
)
A knot diagram
1
Linearized knot diagam
4 5 8 2 10 11 3 12 5 6 8 9
Solving Sequence
6,11 3,7
8 12 10 5 2 4 1 9
c
6
c
7
c
11
c
10
c
5
c
2
c
4
c
1
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.45733 × 10
20
u
26
− 2.14063 × 10
20
u
25
+ ··· + 3.64969 × 10
20
b + 1.19460 × 10
21
,
− 4.51116 × 10
20
u
26
− 7.95319 × 10
20
u
25
+ ··· + 3.64969 × 10
20
a + 5.99946 × 10
21
,
u
27
+ 2u
26
+ ··· − 12u − 4i
I
u
2
= hb − u + 1, −u
2
+ a + 3, u
3
− u
2
− 2u + 1i
I
u
3
= h−au + b + 1, 2a
2
+ au + 2a − 2u − 3, u
2
− 2i
I
v
1
= ha, b − v −2, v
2
+ 3v + 1i
* 4 irreducible components of dim
C
= 0, with total 36 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−1.46 × 10
20
u
26
− 2.14 × 10
20
u
25
+ · · · + 3.65 × 10
20
b + 1.19 ×
10
21
, −4.51 × 10
20
u
26
− 7.95 × 10
20
u
25
+ · · · + 3.65 × 10
20
a + 6.00 ×
10
21
, u
27
+ 2u
26
+ · · · − 12u − 4i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
1.23604u
26
+ 2.17914u
25
+ ··· + 8.25237u − 16.4383
0.399304u
26
+ 0.586525u
25
+ ··· − 3.39246u − 3.27317
a
7
=
1
u
2
a
8
=
0.738918u
26
+ 1.22294u
25
+ ··· + 2.14974u − 8.57454
0.353041u
26
+ 0.631631u
25
+ ··· − 2.16585u − 3.44416
a
12
=
0.175121u
26
+ 0.148966u
25
+ ··· − 7.10327u + 0.964438
0.560998u
26
+ 0.740275u
25
+ ··· − 2.78768u − 4.16594
a
10
=
u
u
a
5
=
−u
2
+ 1
−u
2
a
2
=
1.66235u
26
+ 2.73772u
25
+ ··· + 6.10273u − 20.0632
0.563386u
26
+ 0.613561u
25
+ ··· − 4.52635u − 3.30149
a
4
=
0.670706u
26
+ 1.16883u
25
+ ··· + 9.64782u − 9.46907
−0.635285u
26
− 0.677229u
25
+ ··· + 4.25702u + 3.47590
a
1
=
−0.0973141u
26
+ 0.0651568u
25
+ ··· + 7.52379u − 2.64979
−0.591878u
26
− 0.655073u
25
+ ··· + 3.31129u + 3.50017
a
9
=
−u
3
+ 2u
−u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
237495891437804361646
91242178035245717881
u
26
+
405928781365325849828
91242178035245717881
u
25
+ ··· +
4084274379373058773658
91242178035245717881
u −
5388560215252919173048
91242178035245717881
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
27
− 7u
26
+ ··· − 9u − 1
c
3
, c
7
u
27
− 2u
26
+ ··· − 52u + 8
c
5
, c
6
, c
9
c
10
u
27
− 2u
26
+ ··· − 12u + 4
c
8
, c
11
, c
12
u
27
+ 4u
26
+ ··· − 57u − 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
27
− 15y
26
+ ··· + 103y −1
c
3
, c
7
y
27
+ 12y
26
+ ··· + 3920y −64
c
5
, c
6
, c
9
c
10
y
27
− 24y
26
+ ··· + 432y −16
c
8
, c
11
, c
12
y
27
− 10y
26
+ ··· + 1179y −81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.103210 + 1.061510I
a = 0.243591 + 1.283730I
b = 0.314942 + 0.159811I
5.33632 − 0.36665I −12.32525 − 0.05039I
u = −0.103210 − 1.061510I
a = 0.243591 − 1.283730I
b = 0.314942 − 0.159811I
5.33632 + 0.36665I −12.32525 + 0.05039I
u = −0.903239 + 0.167691I
a = −1.45135 − 0.68497I
b = −0.485625 − 0.865359I
−3.12027 + 0.78467I −18.4367 − 3.3729I
u = −0.903239 − 0.167691I
a = −1.45135 + 0.68497I
b = −0.485625 + 0.865359I
−3.12027 − 0.78467I −18.4367 + 3.3729I
u = 0.337221 + 1.048840I
a = 0.052763 − 1.351640I
b = 0.215415 − 0.194118I
3.59374 − 7.31725I −14.7365 + 5.0472I
u = 0.337221 − 1.048840I
a = 0.052763 + 1.351640I
b = 0.215415 + 0.194118I
3.59374 + 7.31725I −14.7365 − 5.0472I
u = 1.104930 + 0.368271I
a = −0.268980 + 0.135914I
b = 0.714285 − 1.043610I
−3.05220 − 3.96537I −17.7910 + 4.2991I
u = 1.104930 − 0.368271I
a = −0.268980 − 0.135914I
b = 0.714285 + 1.043610I
−3.05220 + 3.96537I −17.7910 − 4.2991I
u = 1.020210 + 0.720557I
a = 0.780740 − 0.476181I
b = 1.045480 − 0.589623I
1.52420 + 1.21028I −14.6857 − 1.0471I
u = 1.020210 − 0.720557I
a = 0.780740 + 0.476181I
b = 1.045480 + 0.589623I
1.52420 − 1.21028I −14.6857 + 1.0471I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.33719
a = 0.790731
b = −0.340332
−14.1221 −16.5250
u = −1.374500 + 0.240422I
a = −0.496066 − 0.228689I
b = −1.169080 − 0.741326I
−6.49373 + 0.53770I −14.7868 − 0.8041I
u = −1.374500 − 0.240422I
a = −0.496066 + 0.228689I
b = −1.169080 + 0.741326I
−6.49373 − 0.53770I −14.7868 + 0.8041I
u = 1.40105
a = −11.6087
b = −22.6375
−8.19904 −208.640
u = −1.295330 + 0.551421I
a = 0.862994 + 0.773421I
b = 1.25669 + 1.17478I
1.64132 + 6.06050I −15.3648 − 4.1353I
u = −1.295330 − 0.551421I
a = 0.862994 − 0.773421I
b = 1.25669 − 1.17478I
1.64132 − 6.06050I −15.3648 + 4.1353I
u = 0.279491 + 0.475963I
a = 0.278696 − 0.748546I
b = −0.794845 + 0.094176I
−0.648673 + 0.468512I −12.96489 + 0.08688I
u = 0.279491 − 0.475963I
a = 0.278696 + 0.748546I
b = −0.794845 − 0.094176I
−0.648673 − 0.468512I −12.96489 − 0.08688I
u = −1.45969
a = −1.05208
b = −1.81866
−6.73578 −12.1710
u = 1.44790 + 0.52184I
a = −0.710802 + 0.417384I
b = −1.42465 + 1.21712I
0.46040 − 5.31882I −15.6377 + 3.3723I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.44790 − 0.52184I
a = −0.710802 − 0.417384I
b = −1.42465 − 1.21712I
0.46040 + 5.31882I −15.6377 − 3.3723I
u = −0.459185
a = 1.01896
b = 1.69587
−10.8656 −30.2970
u = −1.51780 + 0.43184I
a = −0.849204 − 0.755740I
b = −1.65455 − 1.79207I
−2.32586 + 12.67780I −18.7053 − 6.5054I
u = −1.51780 − 0.43184I
a = −0.849204 + 0.755740I
b = −1.65455 + 1.79207I
−2.32586 − 12.67780I −18.7053 + 6.5054I
u = 0.379857
a = 0.653894
b = −0.255787
−0.575852 −17.0320
u = −0.278308
a = −8.72003
b = −0.183695
−2.85525 −50.3790
u = 1.76214
a = 0.0324794
b = −0.496020
−19.5641 −33.0880
7
II. I
u
2
= hb − u + 1, −u
2
+ a + 3, u
3
− u
2
− 2u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
u
2
− 3
u − 1
a
7
=
1
u
2
a
8
=
1
u
2
a
12
=
−u
−u
2
− u + 1
a
10
=
u
u
a
5
=
−u
2
+ 1
−u
2
a
2
=
2u
2
− 4
u
2
+ u − 1
a
4
=
u
2
− 3
u − 1
a
1
=
u
2
− 1
u
2
a
9
=
−u
2
+ 1
−u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
+ 4u − 16
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
7
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
8
u
3
− u
2
− 2u + 1
c
9
, c
10
, c
11
c
12
u
3
+ u
2
− 2u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
7
y
3
c
5
, c
6
, c
8
c
9
, c
10
, c
11
c
12
y
3
− 5y
2
+ 6y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.24698
a = −1.44504
b = −2.24698
−7.98968 −19.4330
u = 0.445042
a = −2.80194
b = −0.554958
−2.34991 −14.0220
u = 1.80194
a = 0.246980
b = 0.801938
−19.2692 −5.54530
11
III. I
u
3
= h−au + b + 1, 2a
2
+ au + 2a − 2u − 3, u
2
− 2i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
a
au − 1
a
7
=
1
2
a
8
=
−
1
2
u
−au + 2a − u + 2
a
12
=
−
1
2
u
−au + 2a + 2
a
10
=
u
u
a
5
=
−1
−2
a
2
=
au − a − 1
3au − 4a − 3
a
4
=
−au + 2a −
1
2
u
−3au + 6a − u + 2
a
1
=
−
1
2
u
−au + 2a − u + 2
a
9
=
0
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −24
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
(u
2
+ u − 1)
2
c
3
, c
4
(u
2
− u − 1)
2
c
5
, c
6
, c
9
c
10
(u
2
− 2)
2
c
8
(u + 1)
4
c
11
, c
12
(u − 1)
4
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
(y
2
− 3y + 1)
2
c
5
, c
6
, c
9
c
10
(y −2)
4
c
8
, c
11
, c
12
(y −1)
4
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = 1.05505
b = 0.492066
−15.4624 −24.0000
u = 1.41421
a = −2.76216
b = −4.90628
−7.56670 −24.0000
u = −1.41421
a = −0.473911
b = −0.329788
−7.56670 −24.0000
u = −1.41421
a = 0.181018
b = −1.25600
−15.4624 −24.0000
15
IV. I
v
1
= ha, b − v − 2, v
2
+ 3v + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
v
0
a
3
=
0
v + 2
a
7
=
1
0
a
8
=
1
v + 3
a
12
=
v − 1
−v − 3
a
10
=
v
0
a
5
=
1
0
a
2
=
v + 2
v + 2
a
4
=
−v − 2
−v − 3
a
1
=
−1
−v − 3
a
9
=
v
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −6
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
u
2
+ u − 1
c
4
, c
7
u
2
− u − 1
c
5
, c
6
, c
9
c
10
u
2
c
8
(u − 1)
2
c
11
, c
12
(u + 1)
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
y
2
− 3y + 1
c
5
, c
6
, c
9
c
10
y
2
c
8
, c
11
, c
12
(y − 1)
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −0.381966
a = 0
b = 1.61803
−10.5276 −6.00000
v = −2.61803
a = 0
b = −0.618034
−2.63189 −6.00000
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
3
)(u
2
+ u − 1)
3
(u
27
− 7u
26
+ ··· − 9u − 1)
c
3
u
3
(u
2
− u − 1)
2
(u
2
+ u − 1)(u
27
− 2u
26
+ ··· − 52u + 8)
c
4
((u + 1)
3
)(u
2
− u − 1)
3
(u
27
− 7u
26
+ ··· − 9u − 1)
c
5
, c
6
u
2
(u
2
− 2)
2
(u
3
− u
2
− 2u + 1)(u
27
− 2u
26
+ ··· − 12u + 4)
c
7
u
3
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
27
− 2u
26
+ ··· − 52u + 8)
c
8
((u − 1)
2
)(u + 1)
4
(u
3
− u
2
− 2u + 1)(u
27
+ 4u
26
+ ··· − 57u − 9)
c
9
, c
10
u
2
(u
2
− 2)
2
(u
3
+ u
2
− 2u − 1)(u
27
− 2u
26
+ ··· − 12u + 4)
c
11
, c
12
((u − 1)
4
)(u + 1)
2
(u
3
+ u
2
− 2u − 1)(u
27
+ 4u
26
+ ··· − 57u − 9)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y − 1)
3
)(y
2
− 3y + 1)
3
(y
27
− 15y
26
+ ··· + 103y − 1)
c
3
, c
7
y
3
(y
2
− 3y + 1)
3
(y
27
+ 12y
26
+ ··· + 3920y − 64)
c
5
, c
6
, c
9
c
10
y
2
(y − 2)
4
(y
3
− 5y
2
+ 6y − 1)(y
27
− 24y
26
+ ··· + 432y − 16)
c
8
, c
11
, c
12
((y − 1)
6
)(y
3
− 5y
2
+ 6y − 1)(y
27
− 10y
26
+ ··· + 1179y − 81)
21
