10
155
(K10n
39
)
A knot diagram
1
Linearized knot diagam
6 7 8 10 9 2 10 5 3 5
Solving Sequence
7,10 5,8
4 3 2 6 1 9
c
7
c
4
c
3
c
2
c
6
c
1
c
9
c
5
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
3
+ 3u
2
+ b + 1, u
3
+ u
2
+ a − u, u
4
+ 3u
3
+ 2u
2
+ 1i
I
u
2
= h−3u
3
+ u
2
+ 2b + u − 8, −2u
3
+ u
2
+ 2a + u − 5, u
4
+ u
3
− u
2
+ 2u + 4i
I
u
3
= hu
2
+ b − 1, u
3
− u
2
+ a − u + 2, u
4
− u
3
− 2u
2
+ 2u + 1i
I
u
4
= h−au + b − 1, a
2
+ au − a + u, u
2
− u − 1i
I
u
5
= h−au + b − u + 2, a
2
− 2au + 3a − 2u + 4, u
2
− u − 1i
* 5 irreducible components of dim
C
= 0, with total 20 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
= h2u
3
+ 3u
2
+ b + 1, u
3
+ u
2
+ a − u, u
4
+ 3u
3
+ 2u
2
+ 1i
(i) Arc colorings
a
7
=
1
0
a
10
=
0
u
a
5
=
−u
3
− u
2
+ u
−2u
3
− 3u
2
− 1
a
8
=
1
u
2
a
4
=
−u
3
− u
2
+ u
−5u
3
− 7u
2
+ u − 3
a
3
=
u
3
+ 2u
2
+ u + 1
−u
a
2
=
u
3
+ 2u
2
+ 1
−u
a
6
=
−u
3
− 2u
2
+ u
−u
2
a
1
=
−u
2
u
3
− u
a
9
=
u
3
+ u
2
+ 1
2u
3
+ 3u
2
+ 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
3
+ 2u
2
+ 10u + 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
u
4
− 3u
3
+ 2u
2
+ 1
c
3
, c
4
, c
10
u
4
+ u
3
+ 5u
2
− u + 1
c
5
, c
8
, c
9
u
4
− 3u
3
+ 5u
2
− 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
y
4
− 5y
3
+ 6y
2
+ 4y + 1
c
3
, c
4
, c
10
y
4
+ 9y
3
+ 29y
2
+ 9y + 1
c
5
, c
8
, c
9
y
4
+ y
3
+ 9y
2
+ y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.192440 + 0.547877I
a = 0.621744 + 0.440597I
b = 0.121744 − 0.425428I
0.204105 − 1.131010I 2.73047 + 6.10768I
u = 0.192440 − 0.547877I
a = 0.621744 − 0.440597I
b = 0.121744 + 0.425428I
0.204105 + 1.131010I 2.73047 − 6.10768I
u = −1.69244 + 0.31815I
a = −0.121744 − 1.306620I
b = −0.62174 − 2.17265I
−13.3636 + 9.2505I −1.73047 − 4.37563I
u = −1.69244 − 0.31815I
a = −0.121744 + 1.306620I
b = −0.62174 + 2.17265I
−13.3636 − 9.2505I −1.73047 + 4.37563I
5
II.
I
u
2
= h−3u
3
+ u
2
+ 2b + u − 8, −2u
3
+ u
2
+ 2a + u − 5, u
4
+ u
3
− u
2
+ 2u + 4i
(i) Arc colorings
a
7
=
1
0
a
10
=
0
u
a
5
=
u
3
−
1
2
u
2
−
1
2
u +
5
2
3
2
u
3
−
1
2
u
2
−
1
2
u + 4
a
8
=
1
u
2
a
4
=
u
3
−
1
2
u
2
−
1
2
u +
5
2
7
2
u
3
−
3
2
u
2
−
3
2
u + 10
a
3
=
−
1
2
u
3
−
3
2
1
2
u
3
−
1
2
u
2
+
1
2
u + 2
a
2
=
−
1
2
u
2
+
1
2
u +
1
2
1
2
u
3
−
1
2
u
2
+
1
2
u + 2
a
6
=
5
4
u
3
−
1
4
u
2
−
3
4
u + 3
3
2
u
3
−
1
2
u
2
−
3
2
u + 3
a
1
=
−
3
4
u
3
−
1
4
u
2
+
5
4
u − 3
−
1
2
u
3
−
1
2
u
2
+
1
2
u − 3
a
9
=
1
4
u
3
−
1
4
u
2
+
1
4
u + 1
1
2
u
3
+
1
2
u
2
+
1
2
u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
3
+ 4u
2
+ 4u − 14
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
u
4
− u
3
− u
2
− 2u + 4
c
3
, c
4
, c
10
u
4
+ 5u
2
+ 1
c
5
, c
8
, c
9
(u
2
+ u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
y
4
− 3y
3
+ 5y
2
− 12y + 16
c
3
, c
4
, c
10
(y
2
+ 5y + 1)
2
c
5
, c
8
, c
9
(y
2
+ y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.895640 + 1.094450I
a = −0.250000 − 0.204588I
b = 0.456850I
−4.93480 − 4.05977I −2.00000 + 6.92820I
u = 0.895640 − 1.094450I
a = −0.250000 + 0.204588I
b = − 0.456850I
−4.93480 + 4.05977I −2.00000 − 6.92820I
u = −1.395640 + 0.228430I
a = −0.25000 + 1.52746I
b = 2.18890I
−4.93480 + 4.05977I −2.00000 − 6.92820I
u = −1.395640 − 0.228430I
a = −0.25000 − 1.52746I
b = − 2.18890I
−4.93480 − 4.05977I −2.00000 + 6.92820I
9
III. I
u
3
= hu
2
+ b − 1, u
3
− u
2
+ a − u + 2, u
4
− u
3
− 2u
2
+ 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
10
=
0
u
a
5
=
−u
3
+ u
2
+ u − 2
−u
2
+ 1
a
8
=
1
u
2
a
4
=
−u
3
+ u
2
+ u − 2
−u
3
− u
2
+ u + 1
a
3
=
−u
3
+ 2u
2
+ u − 3
−u
a
2
=
−u
3
+ 2u
2
− 3
−u
a
6
=
u
3
− 2u
2
− u + 2
−u
2
a
1
=
u
2
− 2
u
3
− u
a
9
=
u
3
− u
2
− 2u + 3
u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6u
3
− 6u
2
− 10u + 9
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
u
4
− u
3
− 2u
2
+ 2u + 1
c
3
, c
10
u
4
− u
3
+ u
2
+ u − 1
c
4
u
4
+ u
3
+ u
2
− u − 1
c
5
, c
9
u
4
− u
3
− u
2
+ u − 1
c
6
u
4
+ u
3
− 2u
2
− 2u + 1
c
8
u
4
+ u
3
− u
2
− u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
y
4
− 5y
3
+ 10y
2
− 8y + 1
c
3
, c
4
, c
10
y
4
+ y
3
+ y
2
− 3y + 1
c
5
, c
8
, c
9
y
4
− 3y
3
+ y
2
+ y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.28879
a = 0.512876
b = −0.660993
0.459232 −0.922080
u = 1.339090 + 0.446630I
a = −0.667076 − 0.670769I
b = −0.593691 − 1.196160I
−5.36351 − 2.52742I −4.35391 + 2.23809I
u = 1.339090 − 0.446630I
a = −0.667076 + 0.670769I
b = −0.593691 + 1.196160I
−5.36351 + 2.52742I −4.35391 − 2.23809I
u = −0.389391
a = −2.17872
b = 0.848375
3.68806 11.6300
13
IV. I
u
4
= h−au + b − 1, a
2
+ au − a + u, u
2
− u − 1i
(i) Arc colorings
a
7
=
1
0
a
10
=
0
u
a
5
=
a
au + 1
a
8
=
1
u + 1
a
4
=
a
2au + a + 1
a
3
=
−au + a − 1
−u
a
2
=
−au + a − u − 1
−u
a
6
=
−a − 2u
−u − 1
a
1
=
a + u + 1
u + 1
a
9
=
−au − u + 1
−au − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
(u
2
+ u − 1)
2
c
3
, c
4
, c
10
u
4
− 2u
3
+ 5u
2
− 4u − 1
c
5
, c
8
, c
9
u
4
− 3u
3
+ 3u
2
+ 2u − 4
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
(y
2
− 3y + 1)
2
c
3
, c
4
, c
10
y
4
+ 6y
3
+ 7y
2
− 26y + 1
c
5
, c
8
, c
9
y
4
− 3y
3
+ 13y
2
− 28y + 16
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −0.319053
b = 1.19719
2.96088 −2.00000
u = −0.618034
a = 1.93709
b = −0.197186
2.96088 −2.00000
u = 1.61803
a = −0.309017 + 1.233910I
b = 0.50000 + 1.99651I
−12.8305 −2.00000
u = 1.61803
a = −0.309017 − 1.233910I
b = 0.50000 − 1.99651I
−12.8305 −2.00000
17
V. I
u
5
= h−au + b − u + 2, a
2
− 2au + 3a − 2u + 4, u
2
− u − 1i
(i) Arc colorings
a
7
=
1
0
a
10
=
0
u
a
5
=
a
au + u − 2
a
8
=
1
u + 1
a
4
=
a
2au + a + u − 2
a
3
=
−au + a − u + 2
u − 1
a
2
=
−au + a + 1
u − 1
a
6
=
−au + 2a − u + 2
u − 2
a
1
=
au − 2a + 2u − 2
−u + 2
a
9
=
au − a + 2u − 3
au − a + 2u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
(u
2
+ u − 1)
2
c
3
, c
4
, c
10
u
4
+ 3u
3
+ 5u
2
+ 6u + 4
c
5
, c
8
, c
9
(u
2
+ u + 1)
2
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
(y
2
− 3y + 1)
2
c
3
, c
4
, c
10
y
4
+ y
3
− 3y
2
+ 4y + 16
c
5
, c
8
, c
9
(y
2
+ y + 1)
2
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −2.11803 + 0.86603I
b = −1.30902 − 0.53523I
−4.93480 −2.00000
u = −0.618034
a = −2.11803 − 0.86603I
b = −1.30902 + 0.53523I
−4.93480 −2.00000
u = 1.61803
a = 0.118034 + 0.866025I
b = −0.19098 + 1.40126I
−4.93480 −2.00000
u = 1.61803
a = 0.118034 − 0.866025I
b = −0.19098 − 1.40126I
−4.93480 −2.00000
21
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
(u
2
+ u − 1)
4
(u
4
− 3u
3
+ 2u
2
+ 1)(u
4
− u
3
− 2u
2
+ 2u + 1)
· (u
4
− u
3
− u
2
− 2u + 4)
c
3
, c
10
(u
4
+ 5u
2
+ 1)(u
4
− 2u
3
+ 5u
2
− 4u − 1)(u
4
− u
3
+ u
2
+ u − 1)
· (u
4
+ u
3
+ 5u
2
− u + 1)(u
4
+ 3u
3
+ 5u
2
+ 6u + 4)
c
4
(u
4
+ 5u
2
+ 1)(u
4
− 2u
3
+ 5u
2
− 4u − 1)(u
4
+ u
3
+ u
2
− u − 1)
· (u
4
+ u
3
+ 5u
2
− u + 1)(u
4
+ 3u
3
+ 5u
2
+ 6u + 4)
c
5
, c
9
(u
2
+ u + 1)
4
(u
4
− 3u
3
+ 3u
2
+ 2u − 4)(u
4
− 3u
3
+ 5u
2
− 3u + 1)
· (u
4
− u
3
− u
2
+ u − 1)
c
6
(u
2
+ u − 1)
4
(u
4
− 3u
3
+ 2u
2
+ 1)(u
4
− u
3
− u
2
− 2u + 4)
· (u
4
+ u
3
− 2u
2
− 2u + 1)
c
8
(u
2
+ u + 1)
4
(u
4
− 3u
3
+ 3u
2
+ 2u − 4)(u
4
− 3u
3
+ 5u
2
− 3u + 1)
· (u
4
+ u
3
− u
2
− u − 1)
22
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
(y
2
− 3y + 1)
4
(y
4
− 5y
3
+ 6y
2
+ 4y + 1)(y
4
− 5y
3
+ 10y
2
− 8y + 1)
· (y
4
− 3y
3
+ 5y
2
− 12y + 16)
c
3
, c
4
, c
10
(y
2
+ 5y + 1)
2
(y
4
+ y
3
− 3y
2
+ 4y + 16)(y
4
+ y
3
+ y
2
− 3y + 1)
· (y
4
+ 6y
3
+ 7y
2
− 26y + 1)(y
4
+ 9y
3
+ 29y
2
+ 9y + 1)
c
5
, c
8
, c
9
(y
2
+ y + 1)
4
(y
4
− 3y
3
+ y
2
+ y + 1)(y
4
− 3y
3
+ 13y
2
− 28y + 16)
· (y
4
+ y
3
+ 9y
2
+ y + 1)
23
