9
38
(K9a
30
)
A knot diagram
1
Linearized knot diagam
7 5 8 2 9 3 1 6 4
Solving Sequence
5,9 3,6
7 2 1 4 8
c
5
c
6
c
2
c
1
c
4
c
8
c
3
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
10
− 2u
9
− 5u
8
− 11u
7
− 14u
6
− 20u
5
− 23u
4
− 16u
3
− 13u
2
+ 4b − 4u + 1,
u
10
+ 2u
9
+ 5u
8
+ 11u
7
+ 10u
6
+ 12u
5
+ 15u
4
− 7u
2
+ 8a − 4u − 9,
u
11
+ u
10
+ 3u
9
+ 6u
8
+ 7u
7
+ 10u
6
+ 11u
5
+ 9u
4
+ 9u
3
+ 3u
2
+ 3u + 1i
I
u
2
= h20020u
17
− 48508u
16
+ ··· + 11959b − 19142, −16736u
17
+ 49970u
16
+ ··· + 11959a − 645,
u
18
− 3u
17
+ ··· − 2u + 1i
I
u
3
= hb + 1, 2a + 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 30 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
10
−2u
9
+· · ·+4b+1, u
10
+2u
9
+· · ·+8a−9, u
11
+u
10
+· · ·+3u+1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
3
=
−
1
8
u
10
−
1
4
u
9
+ ··· +
1
2
u +
9
8
1
4
u
10
+
1
2
u
9
+ ··· + u −
1
4
a
6
=
1
u
2
a
7
=
−
1
16
u
10
−
1
8
u
9
+ ··· −
7
4
u +
1
16
−
1
8
u
10
−
1
4
u
9
+ ··· −
1
2
u +
1
8
a
2
=
1
8
u
10
+
1
4
u
9
+ ··· +
3
2
u +
7
8
1
4
u
10
+
1
2
u
9
+ ··· + u −
1
4
a
1
=
1
16
u
10
+
1
8
u
9
+ ··· +
7
4
u +
15
16
1
8
u
10
+
1
4
u
9
+ ··· +
3
2
u −
1
8
a
4
=
3
8
u
10
+
3
4
u
9
+ ··· +
5
2
u +
13
8
5
4
u
10
+
3
2
u
9
+ ··· + u −
1
4
a
8
=
u
u
3
+ u
a
8
=
u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
27
16
u
10
+
13
8
u
9
−
23
16
u
8
−
41
16
u
7
+
37
8
u
6
+
7
4
u
5
−
13
16
u
4
+ 7u
3
−
3
16
u
2
+
31
4
u −
181
16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
7
c
8
u
11
+ u
10
+ 3u
9
+ 6u
8
+ 7u
7
+ 10u
6
+ 11u
5
+ 9u
4
+ 9u
3
+ 3u
2
+ 3u + 1
c
2
, c
4
u
11
− 3u
9
+ u
8
+ 4u
7
− 2u
6
− u
5
+ 10u
4
− 5u
3
− 16u
2
+ 9u + 4
c
3
u
11
− 3u
10
+ ··· − 6u + 8
c
6
, c
9
2(2u
11
+ u
10
− 3u
8
+ 11u
7
+ 2u
6
− 14u
5
+ 7u
4
+ 6u
3
− 5u
2
+ 1)
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
c
8
y
11
+ 5y
10
+ 11y
9
+ 8y
8
− 5y
7
+ 47y
5
+ 87y
4
+ 73y
3
+ 27y
2
+ 3y −1
c
2
, c
4
y
11
− 6y
10
+ ··· + 209y −16
c
3
y
11
+ 3y
10
+ ··· + 52y −64
c
6
, c
9
4(4y
11
− y
10
+ ··· + 10y −1)
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.361784 + 0.962924I
a = 0.42885 − 1.90311I
b = −1.019420 + 0.904921I
0.53843 + 4.57539I −8.21994 − 7.99945I
u = −0.361784 − 0.962924I
a = 0.42885 + 1.90311I
b = −1.019420 − 0.904921I
0.53843 − 4.57539I −8.21994 + 7.99945I
u = −1.186630 + 0.355210I
a = 0.360998 + 0.003803I
b = 0.988348 + 0.222965I
−3.44203 − 0.72668I −9.61068 + 7.91738I
u = −1.186630 − 0.355210I
a = 0.360998 − 0.003803I
b = 0.988348 − 0.222965I
−3.44203 + 0.72668I −9.61068 − 7.91738I
u = 0.256965 + 0.681325I
a = 0.565680 + 0.993565I
b = −1.41820 − 0.12736I
−1.48009 − 1.36667I −10.72983 + 4.40179I
u = 0.256965 − 0.681325I
a = 0.565680 − 0.993565I
b = −1.41820 + 0.12736I
−1.48009 + 1.36667I −10.72983 − 4.40179I
u = 0.391610 + 1.210140I
a = −0.57189 + 1.31384I
b = 0.308687 − 1.224930I
6.41512 − 6.30680I −3.61485 + 5.61897I
u = 0.391610 − 1.210140I
a = −0.57189 − 1.31384I
b = 0.308687 + 1.224930I
6.41512 + 6.30680I −3.61485 − 5.61897I
u = 0.57851 + 1.29417I
a = −0.05089 − 1.59336I
b = 1.29294 + 0.67490I
3.25113 − 12.93290I −6.73085 + 7.81031I
u = 0.57851 − 1.29417I
a = −0.05089 + 1.59336I
b = 1.29294 − 0.67490I
3.25113 + 12.93290I −6.73085 − 7.81031I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.357337
a = 1.03450
b = −0.304704
−0.695510 −14.4380
6
II. I
u
2
= h20020u
17
− 48508u
16
+ · · · + 11959b − 19142, −16736u
17
+
49970u
16
+ · · · + 11959a − 645, u
18
− 3u
17
+ · · · − 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
3
=
1.39945u
17
− 4.17844u
16
+ ··· + 1.61753u + 0.0539343
−1.67405u
17
+ 4.05619u
16
+ ··· − 1.50280u + 1.60064
a
6
=
1
u
2
a
7
=
−0.146501u
17
+ 0.631491u
16
+ ··· − 2.98386u + 4.22619
−0.318087u
17
+ 1.15194u
16
+ ··· − 0.807425u − 0.186972
a
2
=
−0.274605u
17
− 0.122251u
16
+ ··· + 0.114725u + 1.65457
−1.67405u
17
+ 4.05619u
16
+ ··· − 1.50280u + 1.60064
a
1
=
−3.02350u
17
+ 9.06932u
16
+ ··· − 5.85985u + 4.11447
−1.51769u
17
+ 3.28171u
16
+ ··· + 1.19801u + 0.319257
a
4
=
2.04465u
17
− 5.56234u
16
+ ··· + 1.85467u − 1.18196
−0.318087u
17
+ 1.15194u
16
+ ··· − 0.807425u − 0.186972
a
8
=
u
u
3
+ u
a
8
=
u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
35580
11959
u
17
−
72112
11959
u
16
+ ··· +
57268
11959
u −
174278
11959
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
7
c
8
u
18
− 3u
17
+ ··· − 2u + 1
c
2
, c
4
(u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1)
2
c
3
(u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1)
2
c
6
, c
9
u
18
− 3u
17
+ ··· + 4u + 11
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
c
8
y
18
+ 11y
17
+ ··· + 14y
2
+ 1
c
2
, c
4
(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1)
2
c
3
(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
2
c
6
, c
9
y
18
+ 7y
17
+ ··· + 1260y + 121
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.131255 + 1.025520I
a = 2.21228 − 3.01855I
b = −0.825933
2.09142 −12.65235 + 0.I
u = −0.131255 − 1.025520I
a = 2.21228 + 3.01855I
b = −0.825933
2.09142 −12.65235 + 0.I
u = 1.068960 + 0.157811I
a = 0.330746 + 0.183937I
b = 1.172470 − 0.500383I
−0.30826 + 7.08493I −9.57680 − 5.91335I
u = 1.068960 − 0.157811I
a = 0.330746 − 0.183937I
b = 1.172470 + 0.500383I
−0.30826 − 7.08493I −9.57680 + 5.91335I
u = 0.255037 + 0.861194I
a = 0.31995 + 1.69908I
b = −1.173910 − 0.391555I
−1.08148 − 1.33617I −11.28409 + 0.70175I
u = 0.255037 − 0.861194I
a = 0.31995 − 1.69908I
b = −1.173910 + 0.391555I
−1.08148 + 1.33617I −11.28409 − 0.70175I
u = −0.287150 + 1.197360I
a = −0.077942 − 1.012210I
b = 0.141484 + 0.739668I
2.67293 + 2.45442I −6.32792 − 2.91298I
u = −0.287150 − 1.197360I
a = −0.077942 + 1.012210I
b = 0.141484 − 0.739668I
2.67293 − 2.45442I −6.32792 + 2.91298I
u = 0.605058 + 1.127080I
a = 0.639032 − 1.048120I
b = 0.772920 + 0.510351I
5.07330 − 2.09337I −3.48501 + 4.16283I
u = 0.605058 − 1.127080I
a = 0.639032 + 1.048120I
b = 0.772920 − 0.510351I
5.07330 + 2.09337I −3.48501 − 4.16283I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.658024 + 0.097431I
a = 0.910679 + 0.215358I
b = 0.141484 − 0.739668I
2.67293 − 2.45442I −6.32792 + 2.91298I
u = 0.658024 − 0.097431I
a = 0.910679 − 0.215358I
b = 0.141484 + 0.739668I
2.67293 + 2.45442I −6.32792 − 2.91298I
u = −0.62758 + 1.28014I
a = 0.023182 + 1.259910I
b = 1.172470 − 0.500383I
−0.30826 + 7.08493I −9.57680 − 5.91335I
u = −0.62758 − 1.28014I
a = 0.023182 − 1.259910I
b = 1.172470 + 0.500383I
−0.30826 − 7.08493I −9.57680 + 5.91335I
u = 0.31006 + 1.39846I
a = −0.515395 + 0.355009I
b = 0.772920 − 0.510351I
5.07330 + 2.09337I −3.48501 − 4.16283I
u = 0.31006 − 1.39846I
a = −0.515395 − 0.355009I
b = 0.772920 + 0.510351I
5.07330 − 2.09337I −3.48501 + 4.16283I
u = −0.351155 + 0.305986I
a = 1.157480 − 0.200845I
b = −1.173910 − 0.391555I
−1.08148 − 1.33617I −11.28409 + 0.70175I
u = −0.351155 − 0.305986I
a = 1.157480 + 0.200845I
b = −1.173910 + 0.391555I
−1.08148 + 1.33617I −11.28409 − 0.70175I
11
III. I
u
3
= hb + 1, 2a + 1, u − 1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
1
a
3
=
−0.5
−1
a
6
=
1
1
a
7
=
1.25
1.5
a
2
=
−1.5
−1
a
1
=
−0.25
0.5
a
4
=
−0.5
−1
a
8
=
1
2
a
8
=
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −9.75
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
8
u + 1
c
2
, c
5
, c
7
u − 1
c
3
u
c
6
2(2u + 1)
c
9
2(2u − 1)
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
7
, c
8
y −1
c
3
y
c
6
, c
9
4(4y −1)
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.500000
b = −1.00000
−3.28987 −9.75000
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
8
(u + 1)
· (u
11
+ u
10
+ 3u
9
+ 6u
8
+ 7u
7
+ 10u
6
+ 11u
5
+ 9u
4
+ 9u
3
+ 3u
2
+ 3u + 1)
· (u
18
− 3u
17
+ ··· − 2u + 1)
c
2
(u − 1)(u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1)
2
· (u
11
− 3u
9
+ u
8
+ 4u
7
− 2u
6
− u
5
+ 10u
4
− 5u
3
− 16u
2
+ 9u + 4)
c
3
u(u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1)
2
· (u
11
− 3u
10
+ ··· − 6u + 8)
c
4
(u + 1)(u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1)
2
· (u
11
− 3u
9
+ u
8
+ 4u
7
− 2u
6
− u
5
+ 10u
4
− 5u
3
− 16u
2
+ 9u + 4)
c
5
, c
7
(u − 1)
· (u
11
+ u
10
+ 3u
9
+ 6u
8
+ 7u
7
+ 10u
6
+ 11u
5
+ 9u
4
+ 9u
3
+ 3u
2
+ 3u + 1)
· (u
18
− 3u
17
+ ··· − 2u + 1)
c
6
4(2u + 1)(2u
11
+ u
10
+ ··· − 5u
2
+ 1)
· (u
18
− 3u
17
+ ··· + 4u + 11)
c
9
4(2u − 1)(2u
11
+ u
10
+ ··· − 5u
2
+ 1)
· (u
18
− 3u
17
+ ··· + 4u + 11)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
7
c
8
(y −1)
· (y
11
+ 5y
10
+ 11y
9
+ 8y
8
− 5y
7
+ 47y
5
+ 87y
4
+ 73y
3
+ 27y
2
+ 3y −1)
· (y
18
+ 11y
17
+ ··· + 14y
2
+ 1)
c
2
, c
4
(y −1)(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1)
2
· (y
11
− 6y
10
+ ··· + 209y −16)
c
3
y(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
2
· (y
11
+ 3y
10
+ ··· + 52y −64)
c
6
, c
9
16(4y −1)(4y
11
− y
10
+ ··· + 10y −1)(y
18
+ 7y
17
+ ··· + 1260y + 121)
17
