10
163
(K10n
35
)
A knot diagram
1
Linearized knot diagam
4 5 7 9 10 4 3 1 2 3
Solving Sequence
3,7
4
1,8
9 6 10 5 2
c
3
c
7
c
8
c
6
c
10
c
5
c
2
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
13
+ 5u
12
+ 15u
11
+ 31u
10
+ 50u
9
+ 63u
8
+ 61u
7
+ 42u
6
+ 17u
5
+ u
4
8u
3
9u
2
+ b 8u + 1,
4u
13
25u
12
+ ··· + 5a + 36,
u
14
+ 5u
13
+ 15u
12
+ 30u
11
+ 47u
10
+ 55u
9
+ 48u
8
+ 22u
7
2u
6
17u
5
15u
4
14u
3
4u
2
+ u + 5i
I
u
2
= h−u
3
a u
3
au + 3u
2
+ 3b + a 4u + 1, u
3
a u
2
a + 2u
3
+ a
2
3u
2
+ 2a + 2u + 3, u
4
u
3
+ u
2
+ u + 1i
I
u
3
= h−u
5
+ 2u
4
4u
3
+ 4u
2
+ b 3u + 1, u
4
+ 2u
3
4u
2
+ a + 3u 3, u
6
2u
5
+ 4u
4
4u
3
+ 4u
2
u + 1i
I
u
4
= h−u
3
a + u
2
a au + b + a + u 1, u
3
a + 3u
2
a + a
2
3au u
2
+ u, u
4
2u
3
+ 2u
2
u + 1i
I
v
1
= ha, b + 1, v 1i
* 5 irreducible components of dim
C
= 0, with total 37 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
13
+5u
12
+· · ·+b+1, 4u
13
25u
12
+· · ·+5a+36, u
14
+5u
13
+· · ·+u+5i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u
2
a
1
=
4
5
u
13
+ 5u
12
+ ···
81
5
u
36
5
u
13
5u
12
+ ··· + 8u 1
a
8
=
u
u
a
9
=
11
5
u
13
+ 8u
12
+ ··· +
61
5
u +
51
5
u
13
u
12
+ ··· 14u + 6
a
6
=
u
u
3
+ u
a
10
=
1
5
u
13
+ 2u
11
+ ···
41
5
u
41
5
u
13
5u
12
+ ··· + 8u 1
a
5
=
9
5
u
13
8u
12
+ ··· +
36
5
u +
26
5
u
13
4u
12
+ ··· 6u 9
a
2
=
1
5
u
13
u
12
+ ···
16
5
u
16
5
2u
12
9u
11
+ ··· + 14u + 4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
13
5u
12
+ 2u
11
+ 43u
10
+ 98u
9
+ 192u
8
+ 233u
7
+ 231u
6
+
106u
5
+ 55u
4
27u
3
21u
2
60u 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
14
+ 4u
12
+ ··· 2u + 3
c
2
, c
4
u
14
u
13
+ ··· 3u + 1
c
3
, c
6
, c
7
u
14
5u
13
+ ··· u + 5
c
8
, c
10
u
14
10u
12
+ ··· 2u + 1
c
9
u
14
+ 10u
13
+ ··· + 28u + 5
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
14
+ 8y
13
+ ··· + 62y + 9
c
2
, c
4
y
14
5y
13
+ ··· 13y + 1
c
3
, c
6
, c
7
y
14
+ 5y
13
+ ··· 41y + 25
c
8
, c
10
y
14
20y
13
+ ··· 6y + 1
c
9
y
14
+ 26y
12
+ ··· + 246y + 25
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.269018 + 0.823102I
a = 0.699358 + 0.808665I
b = 0.020522 0.611730I
0.79193 2.01282I 1.55516 + 4.15380I
u = 0.269018 0.823102I
a = 0.699358 0.808665I
b = 0.020522 + 0.611730I
0.79193 + 2.01282I 1.55516 4.15380I
u = 0.809699 + 0.855443I
a = 0.263291 1.389210I
b = 1.74544 + 0.75171I
4.94416 + 4.48113I 10.56248 7.82532I
u = 0.809699 0.855443I
a = 0.263291 + 1.389210I
b = 1.74544 0.75171I
4.94416 4.48113I 10.56248 + 7.82532I
u = 0.752287 + 0.954057I
a = 0.894691 1.015850I
b = 1.66410 0.12170I
4.62410 + 1.43381I 9.01327 + 1.28996I
u = 0.752287 0.954057I
a = 0.894691 + 1.015850I
b = 1.66410 + 0.12170I
4.62410 1.43381I 9.01327 1.28996I
u = 1.104560 + 0.803929I
a = 0.696159 + 0.641405I
b = 1.59147 + 0.10810I
6.35421 6.00703I 6.42492 + 3.68584I
u = 1.104560 0.803929I
a = 0.696159 0.641405I
b = 1.59147 0.10810I
6.35421 + 6.00703I 6.42492 3.68584I
u = 0.633342 + 0.004347I
a = 0.709307 + 0.875694I
b = 0.273616 0.340717I
1.38615 0.45192I 8.23002 + 1.56844I
u = 0.633342 0.004347I
a = 0.709307 0.875694I
b = 0.273616 + 0.340717I
1.38615 + 0.45192I 8.23002 1.56844I
5
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.17524 + 1.43298I
a = 0.361634 + 0.364044I
b = 0.389777 0.088598I
3.73877 3.84212I 7.98139 + 1.57763I
u = 0.17524 1.43298I
a = 0.361634 0.364044I
b = 0.389777 + 0.088598I
3.73877 + 3.84212I 7.98139 1.57763I
u = 0.91106 + 1.12096I
a = 0.60885 + 1.30444I
b = 1.72243 0.67293I
5.3164 + 13.2900I 4.73276 7.55975I
u = 0.91106 1.12096I
a = 0.60885 1.30444I
b = 1.72243 + 0.67293I
5.3164 13.2900I 4.73276 + 7.55975I
6
II. I
u
2
= h−u
3
a u
3
au + 3u
2
+ 3b + a 4u + 1, u
3
a u
2
a + 2u
3
+ a
2
3u
2
+ 2a + 2u + 3, u
4
u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u
2
a
1
=
a
1
3
u
3
a +
1
3
u
3
+ ···
1
3
a
1
3
a
8
=
u
u
a
9
=
1
3
u
3
a
2
3
u
3
+ ···
1
3
a
4
3
1
a
6
=
u
u
3
+ u
a
10
=
1
3
u
3
a +
1
3
u
3
+ ··· +
2
3
a
1
3
1
3
u
3
a +
1
3
u
3
+ ···
1
3
a
1
3
a
5
=
2
3
u
3
a
1
3
u
3
+ ··· +
1
3
a
5
3
1
3
u
3
a +
1
3
u
3
+ ··· +
2
3
a +
2
3
a
2
=
1
3
u
3
a +
1
3
u
3
+ ··· +
2
3
a
1
3
1
3
u
3
a +
2
3
u
3
+ ··· +
1
3
a +
1
3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
+ 12u
2
8u 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
8
2u
7
+ 4u
6
+ 4u
5
+ 3u
4
+ 11u
3
+ 17u
2
+ 12u + 9
c
2
, c
4
u
8
u
7
+ 2u
6
+ 2u
5
+ 4u
4
3u
3
u
2
+ 2u + 3
c
3
, c
6
, c
7
(u
4
+ u
3
+ u
2
u + 1)
2
c
8
, c
10
u
8
+ u
7
2u
5
8u
4
7u
3
+ 13u
2
+ 8u + 3
c
9
(u
4
u
3
+ u
2
+ u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
8
+ 4y
7
+ 38y
6
+ 86y
5
+ 123y
4
43y
3
+ 79y
2
+ 162y + 81
c
2
, c
4
y
8
+ 3y
7
+ 16y
6
+ 4y
5
+ 34y
4
13y
3
+ 37y
2
10y + 9
c
3
, c
6
, c
7
c
9
(y
4
+ y
3
+ 5y
2
+ y + 1)
2
c
8
, c
10
y
8
y
7
12y
6
+ 36y
5
+ 26y
4
225y
3
+ 233y
2
+ 14y + 9
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
1(vol +
1CS) Cusp shape
u = 0.433380 + 0.525827I
a = 0.49562 1.75938I
b = 0.14207 + 1.77290I
0.59615 + 4.68603I 4.70941 10.27938I
u = 0.433380 + 0.525827I
a = 1.87114 + 1.15272I
b = 0.269251 + 0.341177I
0.59615 + 4.68603I 4.70941 10.27938I
u = 0.433380 0.525827I
a = 0.49562 + 1.75938I
b = 0.14207 1.77290I
0.59615 4.68603I 4.70941 + 10.27938I
u = 0.433380 0.525827I
a = 1.87114 1.15272I
b = 0.269251 0.341177I
0.59615 4.68603I 4.70941 + 10.27938I
u = 0.93338 + 1.13249I
a = 0.415178 0.677087I
b = 1.385970 + 0.175069I
3.88602 4.68603I 7.29059 + 10.27938I
u = 0.93338 + 1.13249I
a = 0.78194 + 1.28375I
b = 1.47465 0.63084I
3.88602 4.68603I 7.29059 + 10.27938I
u = 0.93338 1.13249I
a = 0.415178 + 0.677087I
b = 1.385970 0.175069I
3.88602 + 4.68603I 7.29059 10.27938I
u = 0.93338 1.13249I
a = 0.78194 1.28375I
b = 1.47465 + 0.63084I
3.88602 + 4.68603I 7.29059 10.27938I
10
III. I
u
3
= h−u
5
+ 2u
4
4u
3
+ 4u
2
+ b 3u + 1, u
4
+ 2u
3
4u
2
+ a + 3u
3, u
6
2u
5
+ 4u
4
4u
3
+ 4u
2
u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u
2
a
1
=
u
4
2u
3
+ 4u
2
3u + 3
u
5
2u
4
+ 4u
3
4u
2
+ 3u 1
a
8
=
u
u
a
9
=
2u
5
+ 4u
4
7u
3
+ 7u
2
6u
u
5
u
4
+ 2u
3
u
2
+ u + 1
a
6
=
u
u
3
+ u
a
10
=
u
5
u
4
+ 2u
3
+ 2
u
5
2u
4
+ 4u
3
4u
2
+ 3u 1
a
5
=
u
5
+ 2u
4
4u
3
+ 3u
2
2u 1
u
2
u + 1
a
2
=
u
5
u
4
+ u
3
+ u
2
u + 3
u
4
+ 2u
3
3u
2
+ 2u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
5
+ 12u
4
19u
3
+ 23u
2
16u + 6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
6
+ u
5
+ 2u
4
+ u
3
+ u
2
+ 1
c
2
, c
4
u
6
+ u
4
u
3
+ 2u
2
u + 1
c
3
u
6
2u
5
+ 4u
4
4u
3
+ 4u
2
u + 1
c
6
, c
7
u
6
+ 2u
5
+ 4u
4
+ 4u
3
+ 4u
2
+ u + 1
c
8
, c
10
u
6
3u
5
+ 4u
4
5u
3
+ 5u
2
2u + 1
c
9
u
6
+ 3u
5
+ 4u
4
+ u
3
u
2
+ 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
6
+ 3y
5
+ 4y
4
+ 5y
3
+ 5y
2
+ 2y + 1
c
2
, c
4
y
6
+ 2y
5
+ 5y
4
+ 5y
3
+ 4y
2
+ 3y + 1
c
3
, c
6
, c
7
y
6
+ 4y
5
+ 8y
4
+ 14y
3
+ 16y
2
+ 7y + 1
c
8
, c
10
y
6
y
5
4y
4
+ 5y
3
+ 13y
2
+ 6y + 1
c
9
y
6
y
5
+ 8y
4
7y
3
+ 9y
2
2y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
1(vol +
1CS) Cusp shape
u = 0.937424 + 0.916243I
a = 0.469690 + 0.964836I
b = 1.48299 0.38301I
3.99825 3.41127I 5.61730 + 2.91658I
u = 0.937424 0.916243I
a = 0.469690 0.964836I
b = 1.48299 + 0.38301I
3.99825 + 3.41127I 5.61730 2.91658I
u = 0.096993 + 1.308890I
a = 0.272522 + 0.634620I
b = 0.153300 0.549053I
4.36362 4.05299I 4.55288 + 5.52472I
u = 0.096993 1.308890I
a = 0.272522 0.634620I
b = 0.153300 + 0.549053I
4.36362 + 4.05299I 4.55288 5.52472I
u = 0.034417 + 0.580231I
a = 1.80283 1.48709I
b = 0.136288 + 1.137180I
1.27956 + 3.69612I 0.43558 6.39872I
u = 0.034417 0.580231I
a = 1.80283 + 1.48709I
b = 0.136288 1.137180I
1.27956 3.69612I 0.43558 + 6.39872I
14
IV. I
u
4
= h−u
3
a + u
2
a au + b + a + u 1, u
3
a + 3u
2
a + a
2
3au u
2
+
u, u
4
2u
3
+ 2u
2
u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u
2
a
1
=
a
u
3
a u
2
a + au a u + 1
a
8
=
u
u
a
9
=
au + u
2
+ a u
1
a
6
=
u
u
3
+ u
a
10
=
u
3
a u
2
a + au u + 1
u
3
a u
2
a + au a u + 1
a
5
=
u
3
a + u
2
a + u
3
au 2u
2
+ a + 2u 1
u
3
a + u
2
a + u
3
u
2
+ u
a
2
=
u
3
a 2u
2
a + au u + 1
u
2
a u
3
+ u
2
a u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u
3
12u
2
+ 4u 6
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
8
+ u
7
+ 2u
6
8u
5
+ 6u
4
3u
3
+ 9u
2
2u + 1
c
2
, c
4
u
8
4u
5
+ 7u
4
3u
3
+ u
2
2u + 1
c
3
, c
6
, c
7
(u
4
+ 2u
3
+ 2u
2
+ u + 1)
2
c
8
, c
10
u
8
4u
6
+ 2u
5
+ 3u
4
u
3
+ 3u
2
10u + 7
c
9
(u
4
2u
3
+ 2u
2
u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
8
+ 3y
7
+ 32y
6
16y
5
+ 30y
4
+ 71y
3
+ 81y
2
+ 14y + 1
c
2
, c
4
y
8
+ 14y
6
14y
5
+ 27y
4
11y
3
+ 3y
2
2y + 1
c
3
, c
6
, c
7
c
9
(y
4
+ 2y
2
+ 3y + 1)
2
c
8
, c
10
y
8
8y
7
+ 22y
6
22y
5
+ 3y
4
+ y
3
+ 31y
2
58y + 49
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
1(vol +
1CS) Cusp shape
u = 0.070696 + 0.758745I
a = 0.400494 0.005004I
b = 0.921412 0.580396I
1.74699 2.59539I 1.53952 + 0.91892I
u = 0.070696 + 0.758745I
a = 1.22125 + 2.17765I
b = 0.350716 1.044380I
1.74699 2.59539I 1.53952 + 0.91892I
u = 0.070696 0.758745I
a = 0.400494 + 0.005004I
b = 0.921412 + 0.580396I
1.74699 + 2.59539I 1.53952 0.91892I
u = 0.070696 0.758745I
a = 1.22125 2.17765I
b = 0.350716 + 1.044380I
1.74699 + 2.59539I 1.53952 0.91892I
u = 1.070700 + 0.758745I
a = 0.015173 0.960246I
b = 1.201000 + 0.298580I
5.03685 2.59539I 13.53952 + 0.91892I
u = 1.070700 + 0.758745I
a = 0.893428 + 0.534817I
b = 1.77170 0.19130I
5.03685 2.59539I 13.53952 + 0.91892I
u = 1.070700 0.758745I
a = 0.015173 + 0.960246I
b = 1.201000 0.298580I
5.03685 + 2.59539I 13.53952 0.91892I
u = 1.070700 0.758745I
a = 0.893428 0.534817I
b = 1.77170 + 0.19130I
5.03685 + 2.59539I 13.53952 0.91892I
18
V. I
v
1
= ha, b + 1, v 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
1
0
a
4
=
1
0
a
1
=
0
1
a
8
=
1
0
a
9
=
1
1
a
6
=
1
0
a
10
=
1
1
a
5
=
2
1
a
2
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
8
, c
10
u + 1
c
3
, c
6
, c
7
c
9
u
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
8
, c
10
y 1
c
3
, c
6
, c
7
c
9
y
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
1.64493 6.00000
22
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
(u + 1)(u
6
+ u
5
+ 2u
4
+ u
3
+ u
2
+ 1)
· (u
8
2u
7
+ 4u
6
+ 4u
5
+ 3u
4
+ 11u
3
+ 17u
2
+ 12u + 9)
· (u
8
+ u
7
+ 2u
6
8u
5
+ 6u
4
3u
3
+ 9u
2
2u + 1)
· (u
14
+ 4u
12
+ ··· 2u + 3)
c
2
, c
4
(u + 1)(u
6
+ u
4
+ ··· u + 1)(u
8
4u
5
+ ··· 2u + 1)
· (u
8
u
7
+ ··· + 2u + 3)(u
14
u
13
+ ··· 3u + 1)
c
3
u(u
4
+ u
3
+ u
2
u + 1)
2
(u
4
+ 2u
3
+ 2u
2
+ u + 1)
2
· (u
6
2u
5
+ 4u
4
4u
3
+ 4u
2
u + 1)(u
14
5u
13
+ ··· u + 5)
c
6
, c
7
u(u
4
+ u
3
+ u
2
u + 1)
2
(u
4
+ 2u
3
+ 2u
2
+ u + 1)
2
· (u
6
+ 2u
5
+ 4u
4
+ 4u
3
+ 4u
2
+ u + 1)(u
14
5u
13
+ ··· u + 5)
c
8
, c
10
(u + 1)(u
6
3u
5
+ 4u
4
5u
3
+ 5u
2
2u + 1)
· (u
8
4u
6
+ 2u
5
+ 3u
4
u
3
+ 3u
2
10u + 7)
· (u
8
+ u
7
+ ··· + 8u + 3)(u
14
10u
12
+ ··· 2u + 1)
c
9
u(u
4
2u
3
+ 2u
2
u + 1)
2
(u
4
u
3
+ u
2
+ u + 1)
2
· (u
6
+ 3u
5
+ 4u
4
+ u
3
u
2
+ 1)(u
14
+ 10u
13
+ ··· + 28u + 5)
23
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y 1)(y
6
+ 3y
5
+ 4y
4
+ 5y
3
+ 5y
2
+ 2y + 1)
· (y
8
+ 3y
7
+ 32y
6
16y
5
+ 30y
4
+ 71y
3
+ 81y
2
+ 14y + 1)
· (y
8
+ 4y
7
+ 38y
6
+ 86y
5
+ 123y
4
43y
3
+ 79y
2
+ 162y + 81)
· (y
14
+ 8y
13
+ ··· + 62y + 9)
c
2
, c
4
(y 1)(y
6
+ 2y
5
+ 5y
4
+ 5y
3
+ 4y
2
+ 3y + 1)
· (y
8
+ 14y
6
14y
5
+ 27y
4
11y
3
+ 3y
2
2y + 1)
· (y
8
+ 3y
7
+ 16y
6
+ 4y
5
+ 34y
4
13y
3
+ 37y
2
10y + 9)
· (y
14
5y
13
+ ··· 13y + 1)
c
3
, c
6
, c
7
y(y
4
+ 2y
2
+ 3y + 1)
2
(y
4
+ y
3
+ 5y
2
+ y + 1)
2
· (y
6
+ 4y
5
+ ··· + 7y + 1)(y
14
+ 5y
13
+ ··· 41y + 25)
c
8
, c
10
(y 1)(y
6
y
5
4y
4
+ 5y
3
+ 13y
2
+ 6y + 1)
· (y
8
8y
7
+ 22y
6
22y
5
+ 3y
4
+ y
3
+ 31y
2
58y + 49)
· (y
8
y
7
12y
6
+ 36y
5
+ 26y
4
225y
3
+ 233y
2
+ 14y + 9)
· (y
14
20y
13
+ ··· 6y + 1)
c
9
y(y
4
+ 2y
2
+ 3y + 1)
2
(y
4
+ y
3
+ 5y
2
+ y + 1)
2
· (y
6
y
5
+ 8y
4
7y
3
+ 9y
2
2y + 1)(y
14
+ 26y
12
+ ··· + 246y + 25)
24