10
21
(K10a
60
)
A knot diagram
1
Linearized knot diagam
8 7 9 6 10 1 2 3 4 5
Solving Sequence
3,7
2 8 9 4 10 1 6 5
c
2
c
7
c
8
c
3
c
9
c
1
c
6
c
5
c
4
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
16
− u
15
+ ··· − 2u + 1i
I
u
2
= hu
6
+ 2u
4
+ u
3
+ u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 22 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
16
− u
15
+ 7u
14
− 7u
13
+ 20u
12
− 20u
11
+ 27u
10
− 27u
9
+ 12u
8
−
12u
7
− 8u
6
+ 8u
5
− 6u
4
+ 6u
3
+ 3u
2
− 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
2
=
1
−u
2
a
8
=
−u
u
3
+ u
a
9
=
−u
3
− 2u
u
3
+ u
a
4
=
−u
6
− 3u
4
− 2u
2
+ 1
u
6
+ 2u
4
+ u
2
a
10
=
u
9
+ 4u
7
+ 5u
5
− 3u
−u
9
− 3u
7
− 3u
5
+ u
a
1
=
u
2
+ 1
−u
4
− 2u
2
a
6
=
u
5
+ 2u
3
+ u
−u
7
− 3u
5
− 2u
3
+ u
a
5
=
u
15
+ 7u
13
+ ··· − 3u
2
+ u
−u
15
− 7u
13
+ ··· + 3u
2
+ 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
15
− 4u
14
+ 28u
13
− 24u
12
+ 76u
11
− 56u
10
+ 88u
9
− 52u
8
+
8u
7
+ 4u
6
− 64u
5
+ 28u
4
− 24u
3
− 4u
2
+ 20u − 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
u
16
+ u
15
+ ··· + 2u + 1
c
3
, c
6
, c
8
c
9
u
16
+ 2u
15
+ ··· + u + 2
c
4
u
16
+ 9u
15
+ ··· + 2u + 1
c
5
, c
10
u
16
+ u
15
+ ··· + u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
y
16
+ 13y
15
+ ··· + 2y + 1
c
3
, c
6
, c
8
c
9
y
16
− 18y
15
+ ··· + 19y + 4
c
4
y
16
− 3y
15
+ ··· − 2y + 1
c
5
, c
10
y
16
+ 9y
15
+ ··· + 2y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.913611 + 0.024079I
−12.72960 − 4.85972I −13.14726 + 3.11789I
u = 0.913611 − 0.024079I
−12.72960 + 4.85972I −13.14726 − 3.11789I
u = 0.186298 + 1.238560I
2.77285 − 2.45923I −2.72504 + 3.25382I
u = 0.186298 − 1.238560I
2.77285 + 2.45923I −2.72504 − 3.25382I
u = 0.048176 + 1.278470I
4.14984 − 1.95072I −0.93886 + 4.17042I
u = 0.048176 − 1.278470I
4.14984 + 1.95072I −0.93886 − 4.17042I
u = −0.255012 + 1.283570I
0.60263 + 6.60937I −6.51664 − 7.40663I
u = −0.255012 − 1.283570I
0.60263 − 6.60937I −6.51664 + 7.40663I
u = −0.650102 + 0.127920I
−3.75337 + 3.37292I −12.93248 − 5.20888I
u = −0.650102 − 0.127920I
−3.75337 − 3.37292I −12.93248 + 5.20888I
u = −0.427423 + 1.281870I
−4.90049 + 4.73480I −6.47201 − 3.02289I
u = −0.427423 − 1.281870I
−4.90049 − 4.73480I −6.47201 + 3.02289I
u = 0.434047 + 1.303760I
−8.59381 − 9.67514I −9.50822 + 5.97678I
u = 0.434047 − 1.303760I
−8.59381 + 9.67514I −9.50822 − 5.97678I
u = 0.250406 + 0.342321I
−0.577111 − 1.084380I −7.75949 + 5.90127I
u = 0.250406 − 0.342321I
−0.577111 + 1.084380I −7.75949 − 5.90127I
5
II. I
u
2
= hu
6
+ 2u
4
+ u
3
+ u
2
+ u − 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
2
=
1
−u
2
a
8
=
−u
u
3
+ u
a
9
=
−u
3
− 2u
u
3
+ u
a
4
=
−u
4
+ u
3
− u
2
+ u
−u
3
− u + 1
a
10
=
−u
4
− u
2
− 1
−u
3
− u + 1
a
1
=
u
2
+ 1
−u
4
− 2u
2
a
6
=
u
5
+ 2u
3
+ u
−u
5
+ u
4
− u
3
+ u
2
a
5
=
u
3
−u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −10
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
7
, c
10
u
6
+ 2u
4
− u
3
+ u
2
− u − 1
c
3
, c
6
, c
8
c
9
(u
2
− u − 1)
3
c
4
u
6
+ 4u
5
+ 6u
4
+ u
3
− 5u
2
− 3u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
7
, c
10
y
6
+ 4y
5
+ 6y
4
+ y
3
− 5y
2
− 3y + 1
c
3
, c
6
, c
8
c
9
(y
2
− 3y + 1)
3
c
4
y
6
− 4y
5
+ 18y
4
− 35y
3
+ 43y
2
− 19y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.896795
−8.88264 −10.0000
u = −0.248003 + 1.088360I
−0.986960 −10.0000
u = −0.248003 − 1.088360I
−0.986960 −10.0000
u = 0.448397 + 1.266170I
−8.88264 −10.0000
u = 0.448397 − 1.266170I
−8.88264 −10.0000
u = 0.496006
−0.986960 −10.0000
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
(u
6
+ 2u
4
− u
3
+ u
2
− u − 1)(u
16
+ u
15
+ ··· + 2u + 1)
c
3
, c
6
, c
8
c
9
((u
2
− u − 1)
3
)(u
16
+ 2u
15
+ ··· + u + 2)
c
4
(u
6
+ 4u
5
+ 6u
4
+ u
3
− 5u
2
− 3u + 1)(u
16
+ 9u
15
+ ··· + 2u + 1)
c
5
, c
10
(u
6
+ 2u
4
− u
3
+ u
2
− u − 1)(u
16
+ u
15
+ ··· + u
2
+ 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
(y
6
+ 4y
5
+ 6y
4
+ y
3
− 5y
2
− 3y + 1)(y
16
+ 13y
15
+ ··· + 2y + 1)
c
3
, c
6
, c
8
c
9
((y
2
− 3y + 1)
3
)(y
16
− 18y
15
+ ··· + 19y + 4)
c
4
(y
6
− 4y
5
+ ··· − 19y + 1)(y
16
− 3y
15
+ ··· − 2y + 1)
c
5
, c
10
(y
6
+ 4y
5
+ 6y
4
+ y
3
− 5y
2
− 3y + 1)(y
16
+ 9y
15
+ ··· + 2y + 1)
11
