12n
0068
(K12n
0068
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 9 3 5 11 6 12 8 10
Solving Sequence
5,9 3,6
7 10 2 1 4 12 11 8
c
5
c
6
c
9
c
2
c
1
c
4
c
12
c
10
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−3.01211 × 10
31
u
29
+ 4.34016 × 10
31
u
28
+ ··· + 1.97066 × 10
30
b − 8.57936 × 10
32
,
1.09509 × 10
30
u
29
− 1.54101 × 10
30
u
28
+ ··· + 7.03808 × 10
28
a + 2.93651 × 10
31
,
u
30
− 2u
29
+ ··· + 112u − 16i
I
u
2
= hb + 1, −u
8
+ 3u
6
+ u
5
− 4u
4
− 2u
3
+ u
2
+ a + 2u + 1, u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1i
I
v
1
= ha, −v
3
+ 8b − 13, v
4
− 3v
3
+ 8v
2
− 3v + 1i
* 3 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
= h−3.01 × 10
31
u
29
+ 4.34 × 10
31
u
28
+ · · · + 1.97 × 10
30
b − 8.58 ×
10
32
, 1.10 × 10
30
u
29
− 1.54 × 10
30
u
28
+ · · · + 7.04 × 10
28
a + 2.94 ×
10
31
, u
30
− 2u
29
+ · · · + 112u − 16i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
3
=
−15.5596u
29
+ 21.8953u
28
+ ··· + 2227.18u − 417.232
15.2848u
29
− 22.0239u
28
+ ··· − 2273.81u + 435.354
a
6
=
1
u
2
a
7
=
−114.601u
29
+ 164.616u
28
+ ··· + 17005.3u − 3254.00
−12.2418u
29
+ 17.5529u
28
+ ··· + 1815.63u − 347.070
a
10
=
−u
−u
3
+ u
a
2
=
−0.274795u
29
− 0.128633u
28
+ ··· − 46.6332u + 18.1226
15.2848u
29
− 22.0239u
28
+ ··· − 2273.81u + 435.354
a
1
=
−114.601u
29
+ 164.616u
28
+ ··· + 17005.3u − 3254.00
−24.1706u
29
+ 34.7409u
28
+ ··· + 3584.44u − 686.313
a
4
=
80.2622u
29
− 115.883u
28
+ ··· − 12018.0u + 2311.90
28.2365u
29
− 40.6686u
28
+ ··· − 4212.96u + 808.462
a
12
=
−82.1322u
29
+ 117.960u
28
+ ··· + 12185.6u − 2331.50
−46.3428u
29
+ 66.6043u
28
+ ··· + 6876.00u − 1316.29
a
11
=
120.629u
29
− 173.452u
28
+ ··· − 17928.5u + 3434.03
58.6201u
29
− 84.3267u
28
+ ··· − 8714.67u + 1669.08
a
8
=
−102.359u
29
+ 147.063u
28
+ ··· + 15189.6u − 2906.93
−12.2418u
29
+ 17.5529u
28
+ ··· + 1815.63u − 347.070
(ii) Obstruction class = −1
(iii) Cusp Shapes = 18.4364u
29
− 27.2600u
28
+ ··· − 2861.01u + 546.175
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
30
+ 52u
29
+ ··· + 28u + 1
c
2
, c
4
u
30
− 12u
29
+ ··· − 4u − 1
c
3
, c
6
u
30
+ 3u
29
+ ··· − 1024u + 512
c
5
, c
9
u
30
+ 2u
29
+ ··· − 112u − 16
c
7
u
30
− 4u
29
+ ··· + 4u − 1
c
8
, c
11
u
30
− 4u
29
+ ··· + 4u + 1
c
10
, c
12
u
30
− 8u
29
+ ··· + 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
30
− 136y
29
+ ··· + 4192y + 1
c
2
, c
4
y
30
− 52y
29
+ ··· − 28y + 1
c
3
, c
6
y
30
− 63y
29
+ ··· − 1572864y + 262144
c
5
, c
9
y
30
− 30y
29
+ ··· − 2176y + 256
c
7
y
30
− 68y
29
+ ··· − 16y + 1
c
8
, c
11
y
30
+ 8y
29
+ ··· − 4y + 1
c
10
, c
12
y
30
+ 32y
29
+ ··· − 428y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.715139 + 0.446335I
a = 0.637691 + 0.192313I
b = 0.206372 + 0.122164I
1.47077 + 1.88429I −1.09736 − 4.74077I
u = −0.715139 − 0.446335I
a = 0.637691 − 0.192313I
b = 0.206372 − 0.122164I
1.47077 − 1.88429I −1.09736 + 4.74077I
u = −0.520677 + 0.583530I
a = −1.10072 − 2.04738I
b = −1.225610 + 0.025926I
−2.38851 + 1.39225I −9.42086 − 3.19191I
u = −0.520677 − 0.583530I
a = −1.10072 + 2.04738I
b = −1.225610 − 0.025926I
−2.38851 − 1.39225I −9.42086 + 3.19191I
u = −1.297510 + 0.455237I
a = 0.389154 + 0.049150I
b = 0.517584 + 0.471630I
−4.40422 + 6.31187I −8.00000 − 3.70826I
u = −1.297510 − 0.455237I
a = 0.389154 − 0.049150I
b = 0.517584 − 0.471630I
−4.40422 − 6.31187I −8.00000 + 3.70826I
u = 1.350390 + 0.302093I
a = 0.402345 + 0.008016I
b = 0.436591 − 0.605380I
−5.03747 − 0.32171I −10.46137 + 0.I
u = 1.350390 − 0.302093I
a = 0.402345 − 0.008016I
b = 0.436591 + 0.605380I
−5.03747 + 0.32171I −10.46137 + 0.I
u = 0.458152 + 0.404118I
a = 0.901481 − 0.429883I
b = −0.568552 + 0.347997I
−0.690095 + 0.127607I −9.90837 − 0.33008I
u = 0.458152 − 0.404118I
a = 0.901481 + 0.429883I
b = −0.568552 − 0.347997I
−0.690095 − 0.127607I −9.90837 + 0.33008I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.355435 + 0.458702I
a = 0.191023 + 0.000486I
b = 1.63285 − 0.05553I
−8.72787 + 1.60808I −9.05721 + 6.90396I
u = 0.355435 − 0.458702I
a = 0.191023 − 0.000486I
b = 1.63285 + 0.05553I
−8.72787 − 1.60808I −9.05721 − 6.90396I
u = −0.043773 + 0.562236I
a = 1.55055 + 0.58614I
b = −0.101765 − 0.109648I
−0.46641 − 2.28721I −1.63292 + 4.53779I
u = −0.043773 − 0.562236I
a = 1.55055 − 0.58614I
b = −0.101765 + 0.109648I
−0.46641 + 2.28721I −1.63292 − 4.53779I
u = 0.562163 + 0.001137I
a = −4.72549 − 0.32287I
b = −0.951592 + 0.196609I
−1.29017 + 2.42994I −20.9927 + 0.0895I
u = 0.562163 − 0.001137I
a = −4.72549 + 0.32287I
b = −0.951592 − 0.196609I
−1.29017 − 2.42994I −20.9927 − 0.0895I
u = 0.485715
a = 0.919058
b = −0.317479
−0.783101 −12.6230
u = 1.54469 + 0.29004I
a = 1.71848 − 0.49046I
b = 1.98426 + 0.12684I
−13.3728 − 4.6597I 0
u = 1.54469 − 0.29004I
a = 1.71848 + 0.49046I
b = 1.98426 − 0.12684I
−13.3728 + 4.6597I 0
u = 0.12715 + 1.72473I
a = 0.181586 + 0.001643I
b = 2.07079 − 0.05873I
−16.8261 + 3.2961I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.12715 − 1.72473I
a = 0.181586 − 0.001643I
b = 2.07079 + 0.05873I
−16.8261 − 3.2961I 0
u = 1.75662 + 0.28335I
a = −0.969045 + 0.338183I
b = −1.39226 − 0.94810I
−10.34510 − 5.56831I 0
u = 1.75662 − 0.28335I
a = −0.969045 − 0.338183I
b = −1.39226 + 0.94810I
−10.34510 + 5.56831I 0
u = −1.78192
a = 1.50256
b = 2.09847
−17.8492 0
u = −1.78762 + 0.03529I
a = −0.998109 − 0.180287I
b = −1.51752 + 0.83695I
−10.57970 − 1.09876I 0
u = −1.78762 − 0.03529I
a = −0.998109 + 0.180287I
b = −1.51752 − 0.83695I
−10.57970 + 1.09876I 0
u = 1.61551 + 0.87429I
a = 1.026870 − 0.855426I
b = 1.96767 + 0.40947I
18.1572 − 12.2530I 0
u = 1.61551 − 0.87429I
a = 1.026870 + 0.855426I
b = 1.96767 − 0.40947I
18.1572 + 12.2530I 0
u = −1.75729 + 0.77336I
a = 1.083390 + 0.694936I
b = 2.05068 − 0.37214I
16.9360 + 5.5790I 0
u = −1.75729 − 0.77336I
a = 1.083390 − 0.694936I
b = 2.05068 + 0.37214I
16.9360 − 5.5790I 0
7
II. I
u
2
= hb + 1, −u
8
+ 3u
6
+ u
5
− 4u
4
− 2u
3
+ u
2
+ a + 2u + 1, u
9
+ u
8
−
2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
3
=
u
8
− 3u
6
− u
5
+ 4u
4
+ 2u
3
− u
2
− 2u − 1
−1
a
6
=
1
u
2
a
7
=
1
u
2
a
10
=
−u
−u
3
+ u
a
2
=
u
8
− 3u
6
− u
5
+ 4u
4
+ 2u
3
− u
2
− 2u − 2
−1
a
1
=
−1
0
a
4
=
u
8
− 3u
6
− u
5
+ 4u
4
+ 2u
3
− u
2
− 2u − 1
−1
a
12
=
−u
4
+ u
2
− 1
−u
6
+ 2u
4
− u
2
a
11
=
−u
7
+ 2u
5
− 2u
3
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u
3
− 1
a
8
=
−u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
8
− 2u
7
− 2u
6
+ 3u
5
+ 6u
4
− 3u
3
− 3u
2
− 4u − 10
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
9
c
3
, c
6
u
9
c
4
(u + 1)
9
c
5
u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1
c
7
u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1
c
8
u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1
c
9
u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1
c
10
u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1
c
11
u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1
c
12
u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
9
c
3
, c
6
y
9
c
5
, c
9
y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1
c
7
y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1
c
8
, c
11
y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1
c
10
, c
12
y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.772920 + 0.510351I
a = −0.457852 − 1.072010I
b = −1.00000
0.13850 + 2.09337I −8.93344 − 3.71284I
u = −0.772920 − 0.510351I
a = −0.457852 + 1.072010I
b = −1.00000
0.13850 − 2.09337I −8.93344 + 3.71284I
u = 0.825933
a = −1.46592
b = −1.00000
−2.84338 −14.0380
u = 1.173910 + 0.391555I
a = −0.522253 + 0.392004I
b = −1.00000
−6.01628 − 1.33617I −14.5101 + 2.5441I
u = 1.173910 − 0.391555I
a = −0.522253 − 0.392004I
b = −1.00000
−6.01628 + 1.33617I −14.5101 − 2.5441I
u = −0.141484 + 0.739668I
a = 1.63880 − 0.65075I
b = −1.00000
−2.26187 − 2.45442I −7.83172 + 1.00072I
u = −0.141484 − 0.739668I
a = 1.63880 + 0.65075I
b = −1.00000
−2.26187 + 2.45442I −7.83172 − 1.00072I
u = −1.172470 + 0.500383I
a = −0.425734 − 0.444312I
b = −1.00000
−5.24306 + 7.08493I −13.7057 − 8.1735I
u = −1.172470 − 0.500383I
a = −0.425734 + 0.444312I
b = −1.00000
−5.24306 − 7.08493I −13.7057 + 8.1735I
11
III. I
v
1
= ha, −v
3
+ 8b − 13, v
4
− 3v
3
+ 8v
2
− 3v + 1i
(i) Arc colorings
a
5
=
1
0
a
9
=
v
0
a
3
=
0
1
8
v
3
+
13
8
a
6
=
1
0
a
7
=
1
1
8
v
3
+
21
8
a
10
=
v
0
a
2
=
1
8
v
3
+
13
8
1
8
v
3
+
13
8
a
1
=
1
8
v
3
+
13
8
−
1
8
v
3
−
21
8
a
4
=
−
1
8
v
3
−
13
8
−
1
8
v
3
−
21
8
a
12
=
1
4
v
3
+ v +
5
4
−
1
8
v
3
−
21
8
a
11
=
7
8
v
3
− 2v
2
+ 6v −
5
8
−
9
8
v
3
+ 3v
2
− 8v +
3
8
a
8
=
−
1
8
v
3
−
13
8
1
8
v
3
+
21
8
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
9
2
v
3
+ 13v
2
− 33v −
17
2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
− 3u + 1)
2
c
2
, c
3
(u
2
+ u − 1)
2
c
4
, c
6
(u
2
− u − 1)
2
c
5
, c
9
u
4
c
7
(u
2
+ 3u + 1)
2
c
8
, c
12
(u
2
− u + 1)
2
c
10
, c
11
(u
2
+ u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
2
− 7y + 1)
2
c
2
, c
3
, c
4
c
6
(y
2
− 3y + 1)
2
c
5
, c
9
y
4
c
8
, c
10
, c
11
c
12
(y
2
+ y + 1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 0.190983 + 0.330792I
a = 0
b = 1.61803
−8.88264 + 2.02988I −15.5000 − 9.2736I
v = 0.190983 − 0.330792I
a = 0
b = 1.61803
−8.88264 − 2.02988I −15.5000 + 9.2736I
v = 1.30902 + 2.26728I
a = 0
b = −0.618034
−0.98696 + 2.02988I −15.5000 + 2.3454I
v = 1.30902 − 2.26728I
a = 0
b = −0.618034
−0.98696 − 2.02988I −15.5000 − 2.3454I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
2
− 3u + 1)
2
(u
30
+ 52u
29
+ ··· + 28u + 1)
c
2
((u − 1)
9
)(u
2
+ u − 1)
2
(u
30
− 12u
29
+ ··· − 4u − 1)
c
3
u
9
(u
2
+ u − 1)
2
(u
30
+ 3u
29
+ ··· − 1024u + 512)
c
4
((u + 1)
9
)(u
2
− u − 1)
2
(u
30
− 12u
29
+ ··· − 4u − 1)
c
5
u
4
(u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1)
· (u
30
+ 2u
29
+ ··· − 112u − 16)
c
6
u
9
(u
2
− u − 1)
2
(u
30
+ 3u
29
+ ··· − 1024u + 512)
c
7
((u
2
+ 3u + 1)
2
)(u
9
+ 5u
8
+ ··· + u + 1)
· (u
30
− 4u
29
+ ··· + 4u − 1)
c
8
(u
2
− u + 1)
2
(u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1)
· (u
30
− 4u
29
+ ··· + 4u + 1)
c
9
u
4
(u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1)
· (u
30
+ 2u
29
+ ··· − 112u − 16)
c
10
(u
2
+ u + 1)
2
· (u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
· (u
30
− 8u
29
+ ··· + 4u + 1)
c
11
(u
2
+ u + 1)
2
(u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1)
· (u
30
− 4u
29
+ ··· + 4u + 1)
c
12
(u
2
− u + 1)
2
· (u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1)
· (u
30
− 8u
29
+ ··· + 4u + 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
9
)(y
2
− 7y + 1)
2
(y
30
− 136y
29
+ ··· + 4192y + 1)
c
2
, c
4
((y − 1)
9
)(y
2
− 3y + 1)
2
(y
30
− 52y
29
+ ··· − 28y + 1)
c
3
, c
6
y
9
(y
2
− 3y + 1)
2
(y
30
− 63y
29
+ ··· − 1572864y + 262144)
c
5
, c
9
y
4
(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y − 1)
· (y
30
− 30y
29
+ ··· − 2176y + 256)
c
7
(y
2
− 7y + 1)
2
(y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y − 1)
· (y
30
− 68y
29
+ ··· − 16y + 1)
c
8
, c
11
(y
2
+ y + 1)
2
· (y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y − 1)
· (y
30
+ 8y
29
+ ··· − 4y + 1)
c
10
, c
12
((y
2
+ y + 1)
2
)(y
9
+ 7y
8
+ ··· + 13y − 1)
· (y
30
+ 32y
29
+ ··· − 428y + 1)
17
