10
138
(K10n
1
)
A knot diagram
1
Linearized knot diagam
8 4 10 6 8 9 2 5 2 3
Solving Sequence
5,8
6
2,9
10 1 4 3 7
c
5
c
8
c
9
c
1
c
4
c
3
c
7
c
2
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
4
+ u
3
− u
2
+ b + u, −u
4
+ u
3
− 2u
2
+ a + u − 1, u
7
− u
6
+ 3u
5
− 2u
4
+ 3u
3
− 2u
2
− 1i
I
u
2
= hu
13
− 2u
12
+ 6u
11
− 7u
10
+ 11u
9
− 13u
8
+ 14u
7
− 17u
6
+ 10u
5
− 9u
4
+ 7u
3
− 3u
2
+ b + 3u,
− u
12
+ 2u
11
− 5u
10
+ 6u
9
− 9u
8
+ 11u
7
− 12u
6
+ 13u
5
− 8u
4
+ 8u
3
− 5u
2
+ a + 3u − 2,
u
14
− 2u
13
+ 6u
12
− 8u
11
+ 13u
10
− 16u
9
+ 18u
8
− 21u
7
+ 16u
6
− 15u
5
+ 10u
4
− 6u
3
+ 5u
2
− u + 1i
I
u
3
= hb + u, a, u
2
+ u + 1i
I
u
4
= hb + 1, a, u
2
+ u + 1i
* 4 irreducible components of dim
C
= 0, with total 25 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
− u
2
+ b + u, −u
4
+ u
3
− 2u
2
+ a + u − 1, u
7
− u
6
+ 3u
5
−
2u
4
+ 3u
3
− 2u
2
− 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
6
=
1
−u
2
a
2
=
u
4
− u
3
+ 2u
2
− u + 1
u
4
− u
3
+ u
2
− u
a
9
=
u
u
a
10
=
−u
2
− 1
u
5
+ 2u
3
− u
2
+ u − 1
a
1
=
u
4
− u
3
+ 2u
2
− u + 1
−u
6
+ u
5
− u
4
− u
a
4
=
u
2
+ 1
−u
4
a
3
=
−u
3
+ u
2
− u + 1
u
6
+ u
4
− u
3
+ u
2
− u
a
7
=
u
4
+ u
2
+ 1
u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
6
+ 6u
5
− 10u
4
+ 10u
3
− 8u
2
+ 8u − 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
7
+ 5u
6
+ 10u
5
+ 13u
4
+ 18u
3
+ 20u
2
+ 12u + 4
c
2
, c
4
u
7
+ 5u
6
+ 11u
5
+ 10u
4
− u
3
− 8u
2
− 4u − 1
c
3
, c
5
, c
8
c
10
u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ 2u
2
+ 1
c
6
, c
9
u
7
− u
6
− 5u
5
+ 2u
4
+ 7u
3
+ 4u
2
+ 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
7
− 5y
6
+ 6y
5
+ 15y
4
+ 4y
3
− 72y
2
− 16y −16
c
2
, c
4
y
7
− 3y
6
+ 19y
5
− 50y
4
+ 83y
3
− 36y
2
− 1
c
3
, c
5
, c
8
c
10
y
7
+ 5y
6
+ 11y
5
+ 10y
4
− y
3
− 8y
2
− 4y −1
c
6
, c
9
y
7
− 11y
6
+ 43y
5
− 62y
4
+ 15y
3
+ 8y
2
− 4y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.903382
a = 1.65758
b = −0.158515
−3.61413 −1.15360
u = −0.237163 + 1.166790I
a = −0.931299 + 0.562572I
b = −0.626141 + 1.116010I
−4.21141 − 3.35522I −7.88053 + 3.75965I
u = −0.237163 − 1.166790I
a = −0.931299 − 0.562572I
b = −0.626141 − 1.116010I
−4.21141 + 3.35522I −7.88053 − 3.75965I
u = −0.266839 + 0.572668I
a = 0.482335 − 0.961495I
b = −0.260920 − 0.655876I
−0.184850 − 1.357360I −2.08591 + 4.58406I
u = −0.266839 − 0.572668I
a = 0.482335 + 0.961495I
b = −0.260920 + 0.655876I
−0.184850 + 1.357360I −2.08591 − 4.58406I
u = 0.552311 + 1.284990I
a = 0.120172 − 1.321830I
b = 0.46632 − 2.74126I
−11.0685 + 10.4672I −6.45679 − 5.97165I
u = 0.552311 − 1.284990I
a = 0.120172 + 1.321830I
b = 0.46632 + 2.74126I
−11.0685 − 10.4672I −6.45679 + 5.97165I
5
II.
I
u
2
= hu
13
−2u
12
+· · ·+b+3u, −u
12
+2u
11
+· · ·+a−2, u
14
−2u
13
+· · ·−u+1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
6
=
1
−u
2
a
2
=
u
12
− 2u
11
+ ··· − 3u + 2
−u
13
+ 2u
12
+ ··· + 3u
2
− 3u
a
9
=
u
u
a
10
=
−u
2
− 1
u
13
− u
12
+ ··· + u
2
+ 3u
a
1
=
u
12
− 2u
11
+ ··· − 3u + 2
−u
13
+ 3u
12
+ ··· − 4u + 1
a
4
=
u
2
+ 1
−u
4
a
3
=
u
12
− u
11
+ ··· − 2u + 2
−3u
13
+ 6u
12
+ ··· − 5u + 1
a
7
=
u
4
+ u
2
+ 1
u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 5u
13
−8u
12
+25u
11
−27u
10
+45u
9
−53u
8
+56u
7
−68u
6
+41u
5
−40u
4
+30u
3
−14u
2
+15u−5
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
(u
7
− 2u
6
− 3u
5
+ 8u
4
− 2u
3
− 2u
2
− u + 2)
2
c
2
, c
4
u
14
+ 8u
13
+ ··· + 9u + 1
c
3
, c
5
, c
8
c
10
u
14
+ 2u
13
+ ··· + u + 1
c
6
, c
9
u
14
− 2u
13
+ ··· − 5u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
7
− 10y
6
+ 37y
5
− 62y
4
+ 50y
3
− 32y
2
+ 9y −4)
2
c
2
, c
4
y
14
− 4y
13
+ ··· − 15y + 1
c
3
, c
5
, c
8
c
10
y
14
+ 8y
13
+ ··· + 9y + 1
c
6
, c
9
y
14
− 16y
13
+ ··· + 9y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.991355 + 0.114136I
a = −1.71230 − 0.09769I
b = 0.026394 − 0.197164I
−7.46645 − 4.93043I −4.23989 + 2.98386I
u = 0.991355 − 0.114136I
a = −1.71230 + 0.09769I
b = 0.026394 + 0.197164I
−7.46645 + 4.93043I −4.23989 − 2.98386I
u = 0.185175 + 0.946853I
a = −0.600533 + 0.684269I
b = 0.46039 + 1.77594I
−1.11654 + 3.28492I −6.60141 − 2.44171I
u = 0.185175 − 0.946853I
a = −0.600533 − 0.684269I
b = 0.46039 − 1.77594I
−1.11654 − 3.28492I −6.60141 + 2.44171I
u = −0.625804 + 0.953838I
a = 0.688899 + 0.343864I
b = 0.684697 + 0.025265I
−1.11654 − 3.28492I −6.60141 + 2.44171I
u = −0.625804 − 0.953838I
a = 0.688899 − 0.343864I
b = 0.684697 − 0.025265I
−1.11654 + 3.28492I −6.60141 − 2.44171I
u = −0.457566 + 0.656399I
a = 0.143355 − 0.834966I
b = −0.251357 − 0.560891I
−0.165382 − 1.372840I −2.77344 + 4.48022I
u = −0.457566 − 0.656399I
a = 0.143355 + 0.834966I
b = −0.251357 + 0.560891I
−0.165382 + 1.372840I −2.77344 − 4.48022I
u = 0.480471 + 1.270420I
a = −0.237920 + 1.237410I
b = −0.43046 + 2.68133I
−7.46645 + 4.93043I −4.23989 − 2.98386I
u = 0.480471 − 1.270420I
a = −0.237920 − 1.237410I
b = −0.43046 − 2.68133I
−7.46645 − 4.93043I −4.23989 + 2.98386I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.010735 + 0.596013I
a = 0.813209 − 0.794860I
b = −0.506054 − 0.754738I
−0.165382 − 1.372840I −2.77344 + 4.48022I
u = 0.010735 − 0.596013I
a = 0.813209 + 0.794860I
b = −0.506054 + 0.754738I
−0.165382 + 1.372840I −2.77344 − 4.48022I
u = 0.415634 + 1.342520I
a = 0.405289 − 1.309100I
b = 0.51639 − 2.58562I
−12.1121 −7.77053 + 0.I
u = 0.415634 − 1.342520I
a = 0.405289 + 1.309100I
b = 0.51639 + 2.58562I
−12.1121 −7.77053 + 0.I
10
III. I
u
3
= hb + u, a, u
2
+ u + 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
6
=
1
u + 1
a
2
=
0
−u
a
9
=
u
u
a
10
=
u
u + 1
a
1
=
0
−u
a
4
=
−u
−u
a
3
=
1
−u + 1
a
7
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u + 4
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
2
c
2
, c
3
, c
4
c
6
, c
8
, c
9
u
2
− u + 1
c
5
, c
10
u
2
+ u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
2
c
2
, c
3
, c
4
c
5
, c
6
, c
8
c
9
, c
10
y
2
+ y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0
b = 0.500000 − 0.866025I
− 4.05977I 0. + 6.92820I
u = −0.500000 − 0.866025I
a = 0
b = 0.500000 + 0.866025I
4.05977I 0. − 6.92820I
14
IV. I
u
4
= hb + 1, a, u
2
+ u + 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
6
=
1
u + 1
a
2
=
0
−1
a
9
=
u
u
a
10
=
u
2u
a
1
=
0
−1
a
4
=
−u
−u
a
3
=
−u − 1
−u − 2
a
7
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
2
c
2
, c
3
, c
4
c
6
, c
8
, c
9
u
2
− u + 1
c
5
, c
10
u
2
+ u + 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
2
c
2
, c
3
, c
4
c
5
, c
6
, c
8
c
9
, c
10
y
2
+ y + 1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0
b = −1.00000
0 −3.00000
u = −0.500000 − 0.866025I
a = 0
b = −1.00000
0 −3.00000
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
u
4
(u
7
− 2u
6
− 3u
5
+ 8u
4
− 2u
3
− 2u
2
− u + 2)
2
· (u
7
+ 5u
6
+ 10u
5
+ 13u
4
+ 18u
3
+ 20u
2
+ 12u + 4)
c
2
, c
4
(u
2
− u + 1)
2
(u
7
+ 5u
6
+ 11u
5
+ 10u
4
− u
3
− 8u
2
− 4u − 1)
· (u
14
+ 8u
13
+ ··· + 9u + 1)
c
3
, c
8
(u
2
− u + 1)
2
(u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ 2u
2
+ 1)
· (u
14
+ 2u
13
+ ··· + u + 1)
c
5
, c
10
(u
2
+ u + 1)
2
(u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ 2u
2
+ 1)
· (u
14
+ 2u
13
+ ··· + u + 1)
c
6
, c
9
(u
2
− u + 1)
2
(u
7
− u
6
− 5u
5
+ 2u
4
+ 7u
3
+ 4u
2
+ 2u + 1)
· (u
14
− 2u
13
+ ··· − 5u + 1)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
4
(y
7
− 10y
6
+ 37y
5
− 62y
4
+ 50y
3
− 32y
2
+ 9y −4)
2
· (y
7
− 5y
6
+ 6y
5
+ 15y
4
+ 4y
3
− 72y
2
− 16y −16)
c
2
, c
4
(y
2
+ y + 1)
2
(y
7
− 3y
6
+ 19y
5
− 50y
4
+ 83y
3
− 36y
2
− 1)
· (y
14
− 4y
13
+ ··· − 15y + 1)
c
3
, c
5
, c
8
c
10
(y
2
+ y + 1)
2
(y
7
+ 5y
6
+ 11y
5
+ 10y
4
− y
3
− 8y
2
− 4y −1)
· (y
14
+ 8y
13
+ ··· + 9y + 1)
c
6
, c
9
(y
2
+ y + 1)
2
(y
7
− 11y
6
+ 43y
5
− 62y
4
+ 15y
3
+ 8y
2
− 4y −1)
· (y
14
− 16y
13
+ ··· + 9y + 1)
20
