10
77
(K10a
18
)
A knot diagram
1
Linearized knot diagam
4 8 5 2 10 1 9 3 7 6
Solving Sequence
1,7
6 10
3,5
4 9 8 2
c
6
c
10
c
5
c
3
c
9
c
7
c
2
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
22
− u
21
+ ··· + b − 1, u
22
− 9u
20
+ ··· + a − 1, u
23
− 2u
22
+ ··· − u + 1i
I
u
2
= hu
6
− 2u
4
+ u
2
+ b, u
4
− u
2
+ a + 1, u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
+ u
3
− u
2
− 2u − 1i
I
u
3
= hb, a − 1, u + 1i
* 3 irreducible components of dim
C
= 0, with total 33 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
= hu
22
− u
21
+ · · · + b − 1, u
22
− 9u
20
+ · · · + a − 1, u
23
− 2u
22
+ · · · − u + 1i
(i) Arc colorings
a
1
=
0
u
a
7
=
1
0
a
6
=
1
u
2
a
10
=
−u
−u
3
+ u
a
3
=
−u
22
+ 9u
20
+ ··· + 5u + 1
−u
22
+ u
21
+ ··· − u + 1
a
5
=
−u
2
+ 1
−u
4
+ 2u
2
a
4
=
u
22
− u
21
+ ··· + 6u − 1
−u
19
+ 7u
17
+ ··· − 6u
2
− u
a
9
=
u
3
− 2u
−u
3
+ u
a
8
=
u
6
− 3u
4
+ 2u
2
+ 1
−u
6
+ 2u
4
− u
2
a
2
=
−u
21
+ 9u
19
+ ··· + 15u
2
+ 6u
−u
22
+ u
21
+ ··· − u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
21
+ 4u
20
+ 18u
19
− 32u
18
− 70u
17
+ 100u
16
+ 148u
15
− 132u
14
− 164u
13
− 4u
12
+
38u
11
+ 200u
10
+ 130u
9
− 148u
8
− 136u
7
− 68u
6
+ 2u
5
+ 80u
4
+ 50u
3
+ 20u
2
− 6u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
23
− 2u
22
+ ··· + 3u − 1
c
2
, c
8
u
23
− 2u
22
+ ··· + 2u − 2
c
3
u
23
+ 12u
22
+ ··· + 7u + 1
c
5
, c
6
, c
10
u
23
+ 2u
22
+ ··· − u − 1
c
7
, c
9
u
23
− 6u
22
+ ··· + 8u − 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
23
− 12y
22
+ ··· + 7y −1
c
2
, c
8
y
23
− 6y
22
+ ··· + 8y −4
c
3
y
23
+ 32y
21
+ ··· + 31y −1
c
5
, c
6
, c
10
y
23
− 20y
22
+ ··· − 9y −1
c
7
, c
9
y
23
+ 18y
22
+ ··· − 8y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.094963 + 0.875706I
a = 2.34528 + 0.84882I
b = −1.86529 − 0.93050I
−6.35503 − 7.52364I −0.34364 + 6.02284I
u = −0.094963 − 0.875706I
a = 2.34528 − 0.84882I
b = −1.86529 + 0.93050I
−6.35503 + 7.52364I −0.34364 − 6.02284I
u = 0.019170 + 0.819470I
a = 2.62421 − 0.25037I
b = −2.01346 − 0.21505I
−6.84422 + 1.43226I −1.58922 − 0.72835I
u = 0.019170 − 0.819470I
a = 2.62421 + 0.25037I
b = −2.01346 + 0.21505I
−6.84422 − 1.43226I −1.58922 + 0.72835I
u = −1.204480 + 0.336653I
a = 0.431013 − 0.938359I
b = −1.64316 − 0.13209I
0.429871 − 1.292380I 5.93678 + 0.45977I
u = −1.204480 − 0.336653I
a = 0.431013 + 0.938359I
b = −1.64316 + 0.13209I
0.429871 + 1.292380I 5.93678 − 0.45977I
u = −1.261470 + 0.073530I
a = 0.222367 + 0.062621I
b = −0.51599 − 1.45099I
2.49785 − 1.83570I 6.37573 + 3.60335I
u = −1.261470 − 0.073530I
a = 0.222367 − 0.062621I
b = −0.51599 + 1.45099I
2.49785 + 1.83570I 6.37573 − 3.60335I
u = −0.698406
a = −0.537824
b = −0.384144
1.01631 10.3720
u = −0.380828 + 0.580276I
a = 0.191263 − 0.218661I
b = 0.411893 + 0.381927I
0.26922 − 3.59706I 4.75645 + 7.79597I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.380828 − 0.580276I
a = 0.191263 + 0.218661I
b = 0.411893 − 0.381927I
0.26922 + 3.59706I 4.75645 − 7.79597I
u = −1.283800 + 0.366192I
a = −0.89429 + 1.11514I
b = 2.28606 + 0.18751I
−2.78844 − 5.69706I 2.62032 + 4.06061I
u = −1.283800 − 0.366192I
a = −0.89429 − 1.11514I
b = 2.28606 − 0.18751I
−2.78844 + 5.69706I 2.62032 − 4.06061I
u = 1.318900 + 0.354954I
a = 0.95360 + 1.10438I
b = −1.57753 + 1.07523I
1.33811 + 7.00485I 7.04339 − 5.13787I
u = 1.318900 − 0.354954I
a = 0.95360 − 1.10438I
b = −1.57753 − 1.07523I
1.33811 − 7.00485I 7.04339 + 5.13787I
u = 1.369190 + 0.083411I
a = 0.752735 − 0.144610I
b = −0.117460 − 0.451573I
6.97398 + 1.20490I 11.80214 − 0.58796I
u = 1.369190 − 0.083411I
a = 0.752735 + 0.144610I
b = −0.117460 + 0.451573I
6.97398 − 1.20490I 11.80214 + 0.58796I
u = 1.377900 + 0.168105I
a = −0.363007 + 0.227729I
b = −0.140468 + 0.918165I
5.85182 + 6.12354I 9.22962 − 6.59776I
u = 1.377900 − 0.168105I
a = −0.363007 − 0.227729I
b = −0.140468 − 0.918165I
5.85182 − 6.12354I 9.22962 + 6.59776I
u = 1.339590 + 0.393018I
a = −1.38895 − 1.04382I
b = 1.90675 − 1.28425I
−1.85559 + 12.07470I 3.82521 − 8.06520I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.339590 − 0.393018I
a = −1.38895 + 1.04382I
b = 1.90675 + 1.28425I
−1.85559 − 12.07470I 3.82521 + 8.06520I
u = 0.149995 + 0.273260I
a = −0.10530 + 2.51199I
b = 0.460745 − 0.520456I
−1.67067 + 0.60932I −3.84266 − 0.84402I
u = 0.149995 − 0.273260I
a = −0.10530 − 2.51199I
b = 0.460745 + 0.520456I
−1.67067 − 0.60932I −3.84266 + 0.84402I
7
II.
I
u
2
= hu
6
−2u
4
+u
2
+b, u
4
−u
2
+a+1, u
9
−3u
7
−u
6
+3u
5
+2u
4
+u
3
−u
2
−2u−1i
(i) Arc colorings
a
1
=
0
u
a
7
=
1
0
a
6
=
1
u
2
a
10
=
−u
−u
3
+ u
a
3
=
−u
4
+ u
2
− 1
−u
6
+ 2u
4
− u
2
a
5
=
−u
2
+ 1
−u
4
+ 2u
2
a
4
=
−1
−u
2
a
9
=
u
3
− 2u
−u
3
+ u
a
8
=
u
6
− 3u
4
+ 2u
2
+ 1
−u
6
+ 2u
4
− u
2
a
2
=
−u
−u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
6
− 8u
4
− 4u
3
+ 4u
2
+ 4u + 10
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
10
u
9
− 3u
7
+ u
6
+ 3u
5
− 2u
4
+ u
3
+ u
2
− 2u + 1
c
2
, c
8
(u
3
+ u
2
− 1)
3
c
3
u
9
+ 6u
8
+ 15u
7
+ 17u
6
+ 3u
5
− 12u
4
− 9u
3
+ u
2
+ 2u + 1
c
7
, c
9
(u
3
− u
2
+ 2u − 1)
3
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
10
y
9
− 6y
8
+ 15y
7
− 17y
6
+ 3y
5
+ 12y
4
− 9y
3
− y
2
+ 2y −1
c
2
, c
8
(y
3
− y
2
+ 2y −1)
3
c
3
y
9
− 6y
8
+ 27y
7
− 73y
6
+ 139y
5
− 184y
4
+ 83y
3
− 13y
2
+ 2y −1
c
7
, c
9
(y
3
+ 3y
2
+ 2y −1)
3
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.073457 + 0.802780I
a = −2.03355 − 0.26868I
b = 1.66236 + 0.56228I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = −0.073457 − 0.802780I
a = −2.03355 + 0.26868I
b = 1.66236 − 0.56228I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = 1.21243
a = −1.69089
b = −0.324718
1.11345 9.01950
u = −1.180080 + 0.437737I
a = −0.17400 + 1.44838I
b = 1.66236 − 0.56228I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = −1.180080 − 0.437737I
a = −0.17400 − 1.44838I
b = 1.66236 + 0.56228I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = 1.253530 + 0.365043I
a = −0.79245 − 1.71706I
b = 1.66236 − 0.56228I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = 1.253530 − 0.365043I
a = −0.79245 + 1.71706I
b = 1.66236 + 0.56228I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = −0.606217 + 0.320153I
a = −0.654553 − 0.182436I
b = −0.324718
1.11345 9.01951 + 0.I
u = −0.606217 − 0.320153I
a = −0.654553 + 0.182436I
b = −0.324718
1.11345 9.01951 + 0.I
11
III. I
u
3
= hb, a − 1, u + 1i
(i) Arc colorings
a
1
=
0
−1
a
7
=
1
0
a
6
=
1
1
a
10
=
1
0
a
3
=
1
0
a
5
=
0
1
a
4
=
1
1
a
9
=
1
0
a
8
=
1
0
a
2
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
10
u − 1
c
2
, c
7
, c
8
c
9
u
c
4
, c
5
, c
6
u + 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
10
y − 1
c
2
, c
7
, c
8
c
9
y
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.00000
b = 0
0 0
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u
9
− 3u
7
+ u
6
+ 3u
5
− 2u
4
+ u
3
+ u
2
− 2u + 1)
· (u
23
− 2u
22
+ ··· + 3u − 1)
c
2
, c
8
u(u
3
+ u
2
− 1)
3
(u
23
− 2u
22
+ ··· + 2u − 2)
c
3
(u − 1)(u
9
+ 6u
8
+ 15u
7
+ 17u
6
+ 3u
5
− 12u
4
− 9u
3
+ u
2
+ 2u + 1)
· (u
23
+ 12u
22
+ ··· + 7u + 1)
c
4
(u + 1)(u
9
− 3u
7
+ u
6
+ 3u
5
− 2u
4
+ u
3
+ u
2
− 2u + 1)
· (u
23
− 2u
22
+ ··· + 3u − 1)
c
5
, c
6
(u + 1)(u
9
− 3u
7
+ u
6
+ 3u
5
− 2u
4
+ u
3
+ u
2
− 2u + 1)
· (u
23
+ 2u
22
+ ··· − u − 1)
c
7
, c
9
u(u
3
− u
2
+ 2u − 1)
3
(u
23
− 6u
22
+ ··· + 8u − 4)
c
10
(u − 1)(u
9
− 3u
7
+ u
6
+ 3u
5
− 2u
4
+ u
3
+ u
2
− 2u + 1)
· (u
23
+ 2u
22
+ ··· − u − 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y − 1)(y
9
− 6y
8
+ 15y
7
− 17y
6
+ 3y
5
+ 12y
4
− 9y
3
− y
2
+ 2y − 1)
· (y
23
− 12y
22
+ ··· + 7y − 1)
c
2
, c
8
y(y
3
− y
2
+ 2y − 1)
3
(y
23
− 6y
22
+ ··· + 8y − 4)
c
3
(y − 1)
· (y
9
− 6y
8
+ 27y
7
− 73y
6
+ 139y
5
− 184y
4
+ 83y
3
− 13y
2
+ 2y − 1)
· (y
23
+ 32y
21
+ ··· + 31y − 1)
c
5
, c
6
, c
10
(y − 1)(y
9
− 6y
8
+ 15y
7
− 17y
6
+ 3y
5
+ 12y
4
− 9y
3
− y
2
+ 2y − 1)
· (y
23
− 20y
22
+ ··· − 9y − 1)
c
7
, c
9
y(y
3
+ 3y
2
+ 2y − 1)
3
(y
23
+ 18y
22
+ ··· − 8y − 16)
17
