10
61
(K10a
123
)
A knot diagram
1
Linearized knot diagam
5 9 8 1 2 10 4 3 6 7
Solving Sequence
2,9 3,6
10 5 1 8 4 7
c
2
c
9
c
5
c
1
c
8
c
3
c
7
c
4
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
4
− u
3
− 3u
2
+ b − 2u − 1, u
8
+ 3u
7
+ 10u
6
+ 19u
5
+ 31u
4
+ 35u
3
+ 32u
2
+ 2a + 16u + 4,
u
9
+ 3u
8
+ 10u
7
+ 19u
6
+ 31u
5
+ 37u
4
+ 34u
3
+ 22u
2
+ 8u + 2i
I
u
2
= h−u
4
+ u
3
+ au − 3u
2
+ b + 2u − 1, −u
4
a − 2u
4
− 3u
2
a + u
3
+ a
2
− 7u
2
− a + 2u − 3,
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
I
u
3
= hb + 1, 2a − u, u
2
+ 2i
I
v
1
= ha, b − 1, v + 1i
* 4 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
= h−u
4
−u
3
−3u
2
+b−2u−1, u
8
+3u
7
+· · ·+2a+4, u
9
+3u
8
+· · ·+8u+2i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
−u
2
a
6
=
−
1
2
u
8
−
3
2
u
7
+ ··· − 8u − 2
u
4
+ u
3
+ 3u
2
+ 2u + 1
a
10
=
1
2
u
8
+
1
2
u
7
+ ··· + 3u
2
+ u
−u
8
− 2u
7
− 7u
6
− 10u
5
− 15u
4
− 14u
3
− 10u
2
− 3u − 1
a
5
=
−
1
2
u
8
−
3
2
u
7
+ ··· − 6u − 1
u
4
+ u
3
+ 3u
2
+ 2u + 1
a
1
=
−
1
2
u
8
−
3
2
u
7
+ ··· − 7u
2
− 3u
−u
8
− 2u
7
− 7u
6
− 10u
5
− 15u
4
− 14u
3
− 10u
2
− 4u − 1
a
8
=
−u
u
3
+ u
a
4
=
u
2
+ 1
−u
4
− 2u
2
a
7
=
−u
3
− 2u
u
5
+ 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
7
+ 6u
6
+ 18u
5
+ 32u
4
+ 46u
3
+ 48u
2
+ 38u + 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
9
, c
10
u
9
+ u
8
− 6u
7
− 5u
6
+ 12u
5
+ 6u
4
− 8u
3
+ u
2
+ u + 1
c
2
, c
3
, c
7
c
8
u
9
− 3u
8
+ 10u
7
− 19u
6
+ 31u
5
− 37u
4
+ 34u
3
− 22u
2
+ 8u − 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
9
, c
10
y
9
− 13y
8
+ 70y
7
− 197y
6
+ 300y
5
− 232y
4
+ 86y
3
− 29y
2
− y −1
c
2
, c
3
, c
7
c
8
y
9
+ 11y
8
+ 48y
7
+ 105y
6
+ 119y
5
+ 51y
4
− 52y
3
− 88y
2
− 24y −4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.903187
a = −1.73778
b = 1.56954
−9.61991 −8.30450
u = −0.638951 + 0.973621I
a = 0.628534 + 1.228300I
b = −1.59750 − 0.17287I
−12.55800 − 5.12744I −10.43762 + 3.71423I
u = −0.638951 − 0.973621I
a = 0.628534 − 1.228300I
b = −1.59750 + 0.17287I
−12.55800 + 5.12744I −10.43762 − 3.71423I
u = −0.215940 + 0.436674I
a = 0.411654 − 0.740818I
b = 0.234603 + 0.339731I
−0.116751 − 0.880893I −2.67139 + 7.91481I
u = −0.215940 − 0.436674I
a = 0.411654 + 0.740818I
b = 0.234603 − 0.339731I
−0.116751 + 0.880893I −2.67139 − 7.91481I
u = −0.00790 + 1.51466I
a = −0.266916 + 0.385198I
b = −0.581336 − 0.407332I
−6.71646 − 1.46233I −6.34609 + 4.72292I
u = −0.00790 − 1.51466I
a = −0.266916 − 0.385198I
b = −0.581336 + 0.407332I
−6.71646 + 1.46233I −6.34609 − 4.72292I
u = −0.18562 + 1.72176I
a = 0.095616 − 0.974129I
b = 1.65947 + 0.34544I
17.6214 − 8.4586I −11.39264 + 3.44703I
u = −0.18562 − 1.72176I
a = 0.095616 + 0.974129I
b = 1.65947 − 0.34544I
17.6214 + 8.4586I −11.39264 − 3.44703I
5
II. I
u
2
= h−u
4
+ u
3
+ au − 3u
2
+ b + 2u − 1, −u
4
a − 2u
4
+ · · · − a − 3, u
5
−
u
4
+ 4u
3
− 3u
2
+ 3u − 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
−u
2
a
6
=
a
u
4
− u
3
− au + 3u
2
− 2u + 1
a
10
=
−u
4
a + u
3
a − u
4
− 3u
2
a + u
3
+ 2au − 4u
2
− a + 3u − 2
1
a
5
=
u
4
− u
3
− au + 3u
2
+ a − 2u + 1
u
4
− u
3
− au + 3u
2
− 2u + 1
a
1
=
−u
3
a + u
4
+ u
2
a − u
3
− 2au + 4u
2
+ a − 3u + 2
−u
3
a + u
2
a − 2au + a − 1
a
8
=
−u
u
3
+ u
a
4
=
u
2
+ 1
−u
4
− 2u
2
a
7
=
−u
3
− 2u
u
4
− u
3
+ 3u
2
− 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
4
− 4u
3
+ 16u
2
− 12u + 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
9
, c
10
u
10
+ u
9
− 4u
8
− 2u
7
+ 6u
6
− 2u
5
− 7u
4
+ 3u
3
+ 8u
2
+ 2u − 3
c
2
, c
3
, c
7
c
8
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
9
, c
10
y
10
− 9y
9
+ ··· − 52y + 9
c
2
, c
3
, c
7
c
8
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.233677 + 0.885557I
a = −0.504786 − 0.801043I
b = −1.349550 + 0.050168I
−5.10967 + 2.21397I −8.88568 − 4.22289I
u = 0.233677 + 0.885557I
a = −0.32299 + 1.43873I
b = 0.591412 − 0.634202I
−5.10967 + 2.21397I −8.88568 − 4.22289I
u = 0.233677 − 0.885557I
a = −0.504786 + 0.801043I
b = −1.349550 − 0.050168I
−5.10967 − 2.21397I −8.88568 + 4.22289I
u = 0.233677 − 0.885557I
a = −0.32299 − 1.43873I
b = 0.591412 + 0.634202I
−5.10967 − 2.21397I −8.88568 + 4.22289I
u = 0.416284
a = −1.21727
b = 1.15193
−2.40769 −0.391160
u = 0.416284
a = 2.76718
b = −0.506729
−2.40769 −0.391160
u = 0.05818 + 1.69128I
a = −0.032711 − 0.944677I
b = −0.660273 + 1.014190I
−14.2482 + 3.3317I −9.91874 − 2.36228I
u = 0.05818 + 1.69128I
a = 0.585538 + 0.410541I
b = 1.59581 − 0.11029I
−14.2482 + 3.3317I −9.91874 − 2.36228I
u = 0.05818 − 1.69128I
a = −0.032711 + 0.944677I
b = −0.660273 − 1.014190I
−14.2482 − 3.3317I −9.91874 + 2.36228I
u = 0.05818 − 1.69128I
a = 0.585538 − 0.410541I
b = 1.59581 + 0.11029I
−14.2482 − 3.3317I −9.91874 + 2.36228I
9
III. I
u
3
= hb + 1, 2a − u, u
2
+ 2i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
2
a
6
=
1
2
u
−1
a
10
=
1
2
u
u − 1
a
5
=
1
2
u − 1
−1
a
1
=
1
2
u
−1
a
8
=
−u
−u
a
4
=
−1
0
a
7
=
0
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u + 1)
2
c
2
, c
3
, c
7
c
8
u
2
+ 2
c
4
, c
5
, c
9
c
10
(u − 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
9
, c
10
(y − 1)
2
c
2
, c
3
, c
7
c
8
(y + 2)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.414210I
a = 0.707107I
b = −1.00000
−8.22467 −12.0000
u = − 1.414210I
a = − 0.707107I
b = −1.00000
−8.22467 −12.0000
13
IV. I
v
1
= ha, b − 1, v + 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
−1
0
a
3
=
1
0
a
6
=
0
1
a
10
=
−1
−1
a
5
=
1
1
a
1
=
0
−1
a
8
=
−1
0
a
4
=
1
0
a
7
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u − 1
c
2
, c
3
, c
7
c
8
u
c
4
, c
5
, c
9
c
10
u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
9
, c
10
y − 1
c
2
, c
3
, c
7
c
8
y
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 1.00000
−3.28987 −12.0000
17
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
(u − 1)(u + 1)
2
(u
9
+ u
8
+ ··· + u + 1)
· (u
10
+ u
9
− 4u
8
− 2u
7
+ 6u
6
− 2u
5
− 7u
4
+ 3u
3
+ 8u
2
+ 2u − 3)
c
2
, c
3
, c
7
c
8
u(u
2
+ 2)(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
2
· (u
9
− 3u
8
+ 10u
7
− 19u
6
+ 31u
5
− 37u
4
+ 34u
3
− 22u
2
+ 8u − 2)
c
4
, c
5
, c
9
c
10
((u − 1)
2
)(u + 1)(u
9
+ u
8
+ ··· + u + 1)
· (u
10
+ u
9
− 4u
8
− 2u
7
+ 6u
6
− 2u
5
− 7u
4
+ 3u
3
+ 8u
2
+ 2u − 3)
18
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
9
, c
10
(y − 1)
3
· (y
9
− 13y
8
+ 70y
7
− 197y
6
+ 300y
5
− 232y
4
+ 86y
3
− 29y
2
− y − 1)
· (y
10
− 9y
9
+ ··· − 52y + 9)
c
2
, c
3
, c
7
c
8
y(y + 2)
2
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1)
2
· (y
9
+ 11y
8
+ 48y
7
+ 105y
6
+ 119y
5
+ 51y
4
− 52y
3
− 88y
2
− 24y − 4)
19
