10
68
(K10a
67
)
A knot diagram
1
Linearized knot diagam
6 9 8 10 7 2 1 3 5 4
Solving Sequence
2,9 3,6
7 1 5 8 4 10
c
2
c
6
c
1
c
5
c
8
c
3
c
10
c
4
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
11
+ u
10
− 7u
9
+ 6u
8
− 17u
7
+ 12u
6
− 15u
5
+ 7u
4
− 3u
3
− u
2
+ 2b − 3u + 1,
− u
13
+ u
12
− 10u
11
+ 9u
10
− 36u
9
+ 30u
8
− 54u
7
+ 43u
6
− 22u
5
+ 20u
4
+ 10u
3
− 2u
2
+ 4a − 3u + 3,
u
14
+ 9u
12
+ u
11
+ 31u
10
+ 6u
9
+ 48u
8
+ 11u
7
+ 27u
6
+ 2u
5
− 2u
4
− 8u
3
+ u
2
+ 1i
I
u
2
= h4802u
17
− 8268u
16
+ ··· + 12107b + 16224, −1848u
17
− 4160u
16
+ ··· + 12107a − 35011,
u
18
− u
17
+ ··· + 6u + 1i
I
u
3
= h−au + 2b − a − 2u, a
2
+ au + a + 2u, u
2
+ 1i
* 3 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−u
11
+u
10
+· · ·+2b+1, −u
13
+u
12
+· · ·+4a+3, u
14
+9u
12
+· · ·+u
2
+1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
−u
2
a
6
=
1
4
u
13
−
1
4
u
12
+ ··· +
3
4
u −
3
4
1
2
u
11
−
1
2
u
10
+ ··· +
3
2
u −
1
2
a
7
=
1
4
u
13
−
1
4
u
12
+ ··· −
3
4
u −
1
4
1
2
u
11
−
1
2
u
10
+ ··· +
3
2
u −
1
2
a
1
=
u
3
+ 2u
1
4
u
13
−
1
4
u
12
+ ··· +
5
4
u −
1
4
a
5
=
−1
−
1
4
u
13
−
1
4
u
12
+ ··· −
1
4
u −
1
4
a
8
=
−u
u
3
+ u
a
4
=
u
2
+ 1
−u
4
− 2u
2
a
10
=
u
1
4
u
13
−
1
4
u
12
+ ··· +
5
4
u −
1
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
13
−17u
11
−3u
10
−55u
9
−20u
8
−79u
7
−46u
6
−39u
5
−33u
4
+ 9u
3
+ 9u
2
+ 7u + 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
14
− 3u
13
+ ··· − 7u + 2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
u
14
+ 9u
12
+ ··· + u
2
+ 1
c
5
u
14
+ 7u
13
+ ··· + 5u + 4
c
7
u
14
− 9u
13
+ ··· − 115u + 26
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
14
− 7y
13
+ ··· − 5y + 4
c
2
, c
3
, c
4
c
8
, c
9
, c
10
y
14
+ 18y
13
+ ··· + 2y + 1
c
5
y
14
+ y
13
+ ··· + 191y + 16
c
7
y
14
+ 5y
13
+ ··· − 69y + 676
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.552436 + 0.381452I
a = 1.22078 − 1.57866I
b = 1.041840 + 0.481714I
−0.78724 − 4.41668I 3.49417 + 7.88625I
u = −0.552436 − 0.381452I
a = 1.22078 + 1.57866I
b = 1.041840 − 0.481714I
−0.78724 + 4.41668I 3.49417 − 7.88625I
u = −0.04509 + 1.43706I
a = 0.567049 − 0.433483I
b = 0.830389 + 0.784414I
−6.78342 − 2.90589I −2.10855 + 2.91897I
u = −0.04509 − 1.43706I
a = 0.567049 + 0.433483I
b = 0.830389 − 0.784414I
−6.78342 + 2.90589I −2.10855 − 2.91897I
u = 0.498731 + 0.157320I
a = −0.611249 − 0.332083I
b = 0.400528 + 0.482833I
1.035520 + 0.368514I 9.33320 − 2.06000I
u = 0.498731 − 0.157320I
a = −0.611249 + 0.332083I
b = 0.400528 − 0.482833I
1.035520 − 0.368514I 9.33320 + 2.06000I
u = −0.164790 + 0.466680I
a = −1.43454 + 0.30361I
b = −0.941064 + 0.407114I
−1.42730 + 1.54478I 1.163355 − 0.228482I
u = −0.164790 − 0.466680I
a = −1.43454 − 0.30361I
b = −0.941064 − 0.407114I
−1.42730 − 1.54478I 1.163355 + 0.228482I
u = −0.26550 + 1.53094I
a = −0.292054 − 0.268287I
b = 0.243278 − 0.917020I
−10.58650 − 6.18900I −1.00936 + 2.90508I
u = −0.26550 − 1.53094I
a = −0.292054 + 0.268287I
b = 0.243278 + 0.917020I
−10.58650 + 6.18900I −1.00936 − 2.90508I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.33038 + 1.55103I
a = 1.76709 + 0.94504I
b = 1.211210 − 0.579083I
−13.5268 + 11.6370I −3.43423 − 6.31221I
u = 0.33038 − 1.55103I
a = 1.76709 − 0.94504I
b = 1.211210 + 0.579083I
−13.5268 − 11.6370I −3.43423 + 6.31221I
u = 0.19870 + 1.61232I
a = −1.71708 − 0.22802I
b = −1.286170 − 0.280982I
−15.6273 + 2.2414I −5.43859 − 0.46441I
u = 0.19870 − 1.61232I
a = −1.71708 + 0.22802I
b = −1.286170 + 0.280982I
−15.6273 − 2.2414I −5.43859 + 0.46441I
6
II. I
u
2
= h4802u
17
− 8268u
16
+ · · · + 12107b + 16224, −1848u
17
− 4160u
16
+
· · · + 12107a − 35011, u
18
− u
17
+ · · · + 6u + 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
−u
2
a
6
=
0.152639u
17
+ 0.343603u
16
+ ··· + 0.206988u + 2.89180
−0.396630u
17
+ 0.682911u
16
+ ··· − 1.61361u − 1.34005
a
7
=
0.549269u
17
− 0.339308u
16
+ ··· + 1.82060u + 4.23185
−0.396630u
17
+ 0.682911u
16
+ ··· − 1.61361u − 1.34005
a
1
=
0.987528u
17
− 1.29215u
16
+ ··· + 7.85430u + 5.72421
−0.579830u
17
+ 0.616833u
16
+ ··· − 4.42265u − 1.55001
a
5
=
−1.52119u
17
+ 2.03023u
16
+ ··· − 18.2674u − 3.39894
0.275791u
17
− 0.288263u
16
+ ··· + 1.53308u − 0.490956
a
8
=
−u
u
3
+ u
a
4
=
u
2
+ 1
−u
4
− 2u
2
a
10
=
1.50904u
17
− 1.78484u
16
+ ··· + 14.7282u + 7.52119
−0.521516u
17
+ 0.492690u
16
+ ··· − 4.87387u − 1.79698
(ii) Obstruction class = −1
(iii) Cusp Shapes =
31900
12107
u
17
−
61944
12107
u
16
+ ··· +
168364
12107
u +
112870
12107
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
u
18
− u
17
+ ··· + 6u + 1
c
5
(u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1)
2
c
7
(u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
y
18
+ 15y
17
+ ··· − 16y + 1
c
5
(y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1)
2
c
7
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.912264 + 0.491243I
a = −0.78567 − 1.24878I
b = −1.172470 + 0.500383I
−6.88799 + 7.08493I −1.57680 − 5.91335I
u = 0.912264 − 0.491243I
a = −0.78567 + 1.24878I
b = −1.172470 − 0.500383I
−6.88799 − 7.08493I −1.57680 + 5.91335I
u = 0.103396 + 1.069760I
a = −0.757195 − 0.604613I
b = −0.772920 + 0.510351I
−1.50643 + 2.09337I 4.51499 − 4.16283I
u = 0.103396 − 1.069760I
a = −0.757195 + 0.604613I
b = −0.772920 − 0.510351I
−1.50643 − 2.09337I 4.51499 + 4.16283I
u = 0.792965 + 0.741615I
a = 0.617829 − 0.014310I
b = 1.173910 + 0.391555I
−7.66122 − 1.33617I −3.28409 + 0.70175I
u = 0.792965 − 0.741615I
a = 0.617829 + 0.014310I
b = 1.173910 − 0.391555I
−7.66122 + 1.33617I −3.28409 − 0.70175I
u = −0.746849 + 0.515863I
a = 0.408531 − 0.597220I
b = −0.141484 + 0.739668I
−3.90681 − 2.45442I 1.67208 + 2.91298I
u = −0.746849 − 0.515863I
a = 0.408531 + 0.597220I
b = −0.141484 − 0.739668I
−3.90681 + 2.45442I 1.67208 − 2.91298I
u = −0.256179 + 1.094020I
a = 1.04650 − 1.39689I
b = 0.825933
−4.48831 −4.65235 + 0.I
u = −0.256179 − 1.094020I
a = 1.04650 + 1.39689I
b = 0.825933
−4.48831 −4.65235 + 0.I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.118400 + 1.390980I
a = 0.194324 − 0.537825I
b = −0.141484 − 0.739668I
−3.90681 + 2.45442I 1.67208 − 2.91298I
u = 0.118400 − 1.390980I
a = 0.194324 + 0.537825I
b = −0.141484 + 0.739668I
−3.90681 − 2.45442I 1.67208 + 2.91298I
u = 0.00304 + 1.47476I
a = 2.29745 + 0.06492I
b = 1.173910 − 0.391555I
−7.66122 + 1.33617I −3.28409 − 0.70175I
u = 0.00304 − 1.47476I
a = 2.29745 − 0.06492I
b = 1.173910 + 0.391555I
−7.66122 − 1.33617I −3.28409 + 0.70175I
u = −0.18330 + 1.47754I
a = −2.21308 + 0.73195I
b = −1.172470 − 0.500383I
−6.88799 − 7.08493I −1.57680 + 5.91335I
u = −0.18330 − 1.47754I
a = −2.21308 − 0.73195I
b = −1.172470 + 0.500383I
−6.88799 + 7.08493I −1.57680 − 5.91335I
u = −0.243739 + 0.102909I
a = 3.19131 − 0.41254I
b = −0.772920 − 0.510351I
−1.50643 − 2.09337I 4.51499 + 4.16283I
u = −0.243739 − 0.102909I
a = 3.19131 + 0.41254I
b = −0.772920 + 0.510351I
−1.50643 + 2.09337I 4.51499 − 4.16283I
11
III. I
u
3
= h−au + 2b − a − 2u, a
2
+ au + a + 2u, u
2
+ 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
1
a
6
=
a
1
2
au +
1
2
a + u
a
7
=
−
1
2
au +
1
2
a − u
1
2
au +
1
2
a + u
a
1
=
u
−
1
2
au +
1
2
a
a
5
=
−1
1
2
au +
1
2
a
a
8
=
−u
0
a
4
=
0
1
a
10
=
u
−
1
2
au +
1
2
a + u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2au − 2a − 4
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
u
4
− u
2
+ 1
c
2
, c
3
, c
4
c
8
, c
9
, c
10
(u
2
+ 1)
2
c
5
(u
2
− u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
7
(y
2
− y + 1)
2
c
2
, c
3
, c
4
c
8
, c
9
, c
10
(y + 1)
4
c
5
(y
2
+ y + 1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 0.36603 − 1.36603I
b = 0.866025 + 0.500000I
−3.28987 − 2.02988I −2.00000 + 3.46410I
u = 1.000000I
a = −1.36603 + 0.36603I
b = −0.866025 + 0.500000I
−3.28987 + 2.02988I −2.00000 − 3.46410I
u = − 1.000000I
a = 0.36603 + 1.36603I
b = 0.866025 − 0.500000I
−3.28987 + 2.02988I −2.00000 − 3.46410I
u = − 1.000000I
a = −1.36603 − 0.36603I
b = −0.866025 − 0.500000I
−3.28987 − 2.02988I −2.00000 + 3.46410I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
4
− u
2
+ 1)(u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1)
2
· (u
14
− 3u
13
+ ··· − 7u + 2)
c
2
, c
3
, c
4
c
8
, c
9
, c
10
((u
2
+ 1)
2
)(u
14
+ 9u
12
+ ··· + u
2
+ 1)(u
18
− u
17
+ ··· + 6u + 1)
c
5
(u
2
− u + 1)
2
· (u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1)
2
· (u
14
+ 7u
13
+ ··· + 5u + 4)
c
7
(u
4
− u
2
+ 1)
· (u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
2
· (u
14
− 9u
13
+ ··· − 115u + 26)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
2
− y + 1)
2
· (y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1)
2
· (y
14
− 7y
13
+ ··· − 5y + 4)
c
2
, c
3
, c
4
c
8
, c
9
, c
10
((y + 1)
4
)(y
14
+ 18y
13
+ ··· + 2y + 1)(y
18
+ 15y
17
+ ··· − 16y + 1)
c
5
(y
2
+ y + 1)
2
(y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1)
2
· (y
14
+ y
13
+ ··· + 191y + 16)
c
7
(y
2
− y + 1)
2
· (y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
2
· (y
14
+ 5y
13
+ ··· − 69y + 676)
17
