10
70
(K10a
22
)
A knot diagram
1
Linearized knot diagam
8 6 7 10 4 3 9 1 5 2
Solving Sequence
2,6
3 7
4,8
1 9 5 10
c
2
c
6
c
3
c
1
c
8
c
5
c
10
c
4
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
34
+ 2u
33
+ ··· + 2b + 1, u
12
− 5u
10
− 2u
9
+ 9u
8
+ 8u
7
− 4u
6
− 10u
5
− 6u
4
+ 2u
3
+ 5u
2
+ a + 2u + 1,
u
35
− 3u
34
+ ··· + u + 1i
I
u
2
= hb
2
− b + 1, a + 1, u + 1i
* 2 irreducible components of dim
C
= 0, with total 37 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
34
+2u
33
+· · ·+2b+1, u
12
−5u
10
+· · ·+a+1, u
35
−3u
34
+· · ·+u+1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
−u
2
a
7
=
u
−u
3
+ u
a
4
=
−u
2
+ 1
u
4
− 2u
2
a
8
=
−u
12
+ 5u
10
+ ··· − 2u − 1
1
2
u
34
− u
33
+ ··· + u −
1
2
a
1
=
−
1
2
u
34
+ u
33
+ ··· − u +
3
2
−
5
2
u
34
+ 4u
33
+ ··· + 2u +
3
2
a
9
=
−2u
34
+ 3u
33
+ ··· + 6u + 1
7
2
u
34
− 6u
33
+ ··· − 3u −
5
2
a
5
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
10
=
−3u
34
+ 5u
33
+ ··· + u + 3
−
5
2
u
34
+ 4u
33
+ ··· + 2u +
3
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −7u
34
+ 7u
33
+ 100u
32
− 67u
31
− 665u
30
+ 193u
29
+ 2656u
28
+ 332u
27
− 6763u
26
−
4013u
25
+ 10334u
24
+ 13042u
23
− 5838u
22
− 22006u
21
− 10452u
20
+ 17336u
19
+
25367u
18
+ 3521u
17
−19332u
16
−18920u
15
−1888u
14
+ 11508u
13
+ 11034u
12
+ 2874u
11
−
3024u
10
− 4152u
9
− 2538u
8
− 542u
7
+ 450u
6
+ 658u
5
+ 514u
4
+ 230u
3
+ 79u
2
+ 25u + 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
35
+ 2u
34
+ ··· − 2u
2
+ 1
c
2
, c
3
, c
6
u
35
+ 3u
34
+ ··· + u − 1
c
4
, c
9
u
35
+ u
34
+ ··· − 8u − 4
c
5
u
35
− 15u
34
+ ··· − 72u + 16
c
7
, c
10
u
35
+ 12u
34
+ ··· + 4u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
35
+ 12y
34
+ ··· + 4y −1
c
2
, c
3
, c
6
y
35
− 31y
34
+ ··· − 17y −1
c
4
, c
9
y
35
+ 15y
34
+ ··· − 72y −16
c
5
y
35
+ 7y
34
+ ··· − 2016y −256
c
7
, c
10
y
35
+ 24y
34
+ ··· + 40y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.827242 + 0.510777I
a = −1.257960 − 0.317928I
b = 0.711723 − 0.774742I
3.61521 − 1.86508I 8.01949 + 2.70414I
u = −0.827242 − 0.510777I
a = −1.257960 + 0.317928I
b = 0.711723 + 0.774742I
3.61521 + 1.86508I 8.01949 − 2.70414I
u = −0.943343 + 0.501099I
a = −0.582372 − 0.149507I
b = 0.684104 + 0.942114I
3.09693 + 3.49535I 6.37889 − 3.75014I
u = −0.943343 − 0.501099I
a = −0.582372 + 0.149507I
b = 0.684104 − 0.942114I
3.09693 − 3.49535I 6.37889 + 3.75014I
u = −0.253334 + 0.839514I
a = −1.18782 − 1.13183I
b = 0.696750 − 1.005540I
0.97304 − 8.24742I 2.56945 + 7.59916I
u = −0.253334 − 0.839514I
a = −1.18782 + 1.13183I
b = 0.696750 + 1.005540I
0.97304 + 8.24742I 2.56945 − 7.59916I
u = −1.15725
a = −1.05692
b = 0.346138
2.21114 4.02480
u = −0.295449 + 0.784598I
a = −0.058917 − 0.230488I
b = 0.766564 + 0.673327I
1.97084 − 2.68874I 4.58889 + 2.89622I
u = −0.295449 − 0.784598I
a = −0.058917 + 0.230488I
b = 0.766564 − 0.673327I
1.97084 + 2.68874I 4.58889 − 2.89622I
u = −1.164960 + 0.288871I
a = −1.174190 − 0.528323I
b = 0.051770 − 0.955164I
−0.612022 − 1.167710I −0.594633 + 0.482422I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.164960 − 0.288871I
a = −1.174190 + 0.528323I
b = 0.051770 + 0.955164I
−0.612022 + 1.167710I −0.594633 − 0.482422I
u = −0.098834 + 0.725130I
a = −0.03188 + 1.73645I
b = −0.071862 + 1.038610I
−3.83291 − 2.53588I −3.84686 + 3.83326I
u = −0.098834 − 0.725130I
a = −0.03188 − 1.73645I
b = −0.071862 − 1.038610I
−3.83291 + 2.53588I −3.84686 − 3.83326I
u = 1.275860 + 0.152636I
a = 0.756171 + 0.131779I
b = −0.493777 + 1.054750I
2.76473 − 0.81126I 6.02594 + 0.I
u = 1.275860 − 0.152636I
a = 0.756171 − 0.131779I
b = −0.493777 − 1.054750I
2.76473 + 0.81126I 6.02594 + 0.I
u = −1.343360 + 0.175547I
a = 1.08136 + 1.66784I
b = −0.750068 − 0.725396I
4.76978 − 0.62379I 6.88558 + 0.I
u = −1.343360 − 0.175547I
a = 1.08136 − 1.66784I
b = −0.750068 + 0.725396I
4.76978 + 0.62379I 6.88558 + 0.I
u = 1.328700 + 0.290772I
a = 0.892510 − 0.521672I
b = −0.144398 − 1.112500I
0.65547 + 6.20108I 1.95124 − 5.89177I
u = 1.328700 − 0.290772I
a = 0.892510 + 0.521672I
b = −0.144398 + 1.112500I
0.65547 − 6.20108I 1.95124 + 5.89177I
u = −1.349650 + 0.231790I
a = 2.67880 − 0.00517I
b = −0.701280 + 0.976265I
4.00753 − 6.15318I 5.27676 + 5.00692I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.349650 − 0.231790I
a = 2.67880 + 0.00517I
b = −0.701280 − 0.976265I
4.00753 + 6.15318I 5.27676 − 5.00692I
u = 1.360060 + 0.198169I
a = 0.993954 − 0.073655I
b = −0.734023 − 0.241674I
5.16768 + 3.59908I 8.99233 − 3.96847I
u = 1.360060 − 0.198169I
a = 0.993954 + 0.073655I
b = −0.734023 + 0.241674I
5.16768 − 3.59908I 8.99233 + 3.96847I
u = 0.130391 + 0.566931I
a = 1.74005 − 1.54748I
b = −0.611964 − 0.968100I
−0.69789 + 3.19845I −1.06265 − 3.08489I
u = 0.130391 − 0.566931I
a = 1.74005 + 1.54748I
b = −0.611964 + 0.968100I
−0.69789 − 3.19845I −1.06265 + 3.08489I
u = 1.42263 + 0.31147I
a = −0.746326 + 1.154990I
b = 0.845304 − 0.658411I
7.44255 + 6.65019I 0
u = 1.42263 − 0.31147I
a = −0.746326 − 1.154990I
b = 0.845304 + 0.658411I
7.44255 − 6.65019I 0
u = 1.41674 + 0.34279I
a = −2.20298 + 0.34664I
b = 0.724315 + 1.038040I
6.28512 + 12.51090I 0. − 8.16035I
u = 1.41674 − 0.34279I
a = −2.20298 − 0.34664I
b = 0.724315 − 1.038040I
6.28512 − 12.51090I 0. + 8.16035I
u = 1.49697 + 0.02263I
a = −1.77693 − 0.78513I
b = 0.807430 + 0.880445I
11.46670 + 3.01120I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.49697 − 0.02263I
a = −1.77693 + 0.78513I
b = 0.807430 − 0.880445I
11.46670 − 3.01120I 0
u = −0.223261 + 0.425121I
a = −0.321375 + 0.194137I
b = −0.417087 + 0.308331I
0.236326 − 1.154630I 3.51275 + 5.51426I
u = −0.223261 − 0.425121I
a = −0.321375 − 0.194137I
b = −0.417087 − 0.308331I
0.236326 + 1.154630I 3.51275 − 5.51426I
u = 0.146719 + 0.318162I
a = −0.77364 − 1.20062I
b = −0.536572 + 0.742317I
0.11091 − 1.46996I −0.94917 + 3.34118I
u = 0.146719 − 0.318162I
a = −0.77364 + 1.20062I
b = −0.536572 − 0.742317I
0.11091 + 1.46996I −0.94917 − 3.34118I
8
II. I
u
2
= hb
2
− b + 1, a + 1, u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
−1
a
3
=
1
−1
a
7
=
−1
0
a
4
=
0
−1
a
8
=
−1
b
a
1
=
−b + 1
b − 1
a
9
=
0
b − 1
a
5
=
0
−1
a
10
=
0
b − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4b + 5
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
10
u
2
− u + 1
c
2
, c
3
(u + 1)
2
c
4
, c
5
, c
9
u
2
c
6
(u − 1)
2
c
8
u
2
+ u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
8
c
10
y
2
+ y + 1
c
2
, c
3
, c
6
(y −1)
2
c
4
, c
5
, c
9
y
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 3.00000 − 3.46410I
u = −1.00000
a = −1.00000
b = 0.500000 − 0.866025I
1.64493 − 2.02988I 3.00000 + 3.46410I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)(u
35
+ 2u
34
+ ··· − 2u
2
+ 1)
c
2
, c
3
((u + 1)
2
)(u
35
+ 3u
34
+ ··· + u − 1)
c
4
, c
9
u
2
(u
35
+ u
34
+ ··· − 8u − 4)
c
5
u
2
(u
35
− 15u
34
+ ··· − 72u + 16)
c
6
((u − 1)
2
)(u
35
+ 3u
34
+ ··· + u − 1)
c
7
, c
10
(u
2
− u + 1)(u
35
+ 12u
34
+ ··· + 4u − 1)
c
8
(u
2
+ u + 1)(u
35
+ 2u
34
+ ··· − 2u
2
+ 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y
2
+ y + 1)(y
35
+ 12y
34
+ ··· + 4y −1)
c
2
, c
3
, c
6
((y −1)
2
)(y
35
− 31y
34
+ ··· − 17y −1)
c
4
, c
9
y
2
(y
35
+ 15y
34
+ ··· − 72y −16)
c
5
y
2
(y
35
+ 7y
34
+ ··· − 2016y −256)
c
7
, c
10
(y
2
+ y + 1)(y
35
+ 24y
34
+ ··· + 40y −1)
14
