10
131
(K10n
19
)
A knot diagram
1
Linearized knot diagam
4 8 5 2 8 10 1 2 7 6
Solving Sequence
1,4
2
5,8
6 9 3 7 10
c
1
c
4
c
5
c
8
c
3
c
7
c
10
c
2
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−5u
17
− 21u
16
+ ··· + 4b + 15, −15u
17
− 51u
16
+ ··· + 4a + 25, u
18
+ 4u
17
+ ··· − 3u − 1i
I
u
2
= hb − a, a
3
− a
2
+ 1, u − 1i
* 2 irreducible components of dim
C
= 0, with total 21 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−5u
17
− 21u
16
+ · · · + 4b + 15, −15u
17
− 51u
16
+ · · · + 4a +
25, u
18
+ 4u
17
+ · · · − 3u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
8
=
15
4
u
17
+
51
4
u
16
+ ··· − 7u −
25
4
5
4
u
17
+
21
4
u
16
+ ··· − 4u −
15
4
a
6
=
−
1
4
u
17
−
3
4
u
16
+ ··· −
3
2
u +
5
4
−
1
4
u
17
−
3
4
u
16
+ ··· +
1
2
u +
1
4
a
9
=
17
4
u
17
+
57
4
u
16
+ ··· − 8u −
31
4
11
4
u
17
+
35
4
u
16
+ ··· − 5u −
17
4
a
3
=
u
3
u
5
− u
3
+ u
a
7
=
5u
17
+ 18u
16
+ ··· − 11u − 10
5
4
u
17
+
21
4
u
16
+ ··· − 4u −
15
4
a
10
=
11
4
u
17
+
35
4
u
16
+ ··· − 6u −
9
4
2u
17
+
13
2
u
16
+ ··· −
9
2
u −
5
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
17
+
23
2
u
16
+ 15u
15
−
33
2
u
14
−
127
2
u
13
−
91
2
u
12
+ 68u
11
+
110u
10
+
11
2
u
9
−
175
2
u
8
+ 2u
7
+ 53u
6
− 27u
5
− 75u
4
+ 14u
3
+ 41u
2
−
21
2
u −
29
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
18
− 4u
17
+ ··· + 3u − 1
c
2
, c
8
u
18
− u
17
+ ··· − 4u + 8
c
3
u
18
+ 4u
17
+ ··· + 11u + 1
c
5
u
18
− 2u
17
+ ··· − 5u
2
+ 1
c
6
, c
9
, c
10
u
18
− 2u
17
+ ··· + 2u − 1
c
7
u
18
+ 2u
17
+ ··· + 18u − 17
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
18
− 4y
17
+ ··· − 11y + 1
c
2
, c
8
y
18
− 21y
17
+ ··· − 592y + 64
c
3
y
18
+ 24y
17
+ ··· − 11y + 1
c
5
y
18
+ 22y
17
+ ··· − 10y + 1
c
6
, c
9
, c
10
y
18
+ 18y
17
+ ··· − 10y + 1
c
7
y
18
+ 10y
17
+ ··· − 1106y + 289
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.10588
a = 0.709778
b = 0.371475
−2.12974 −1.01840
u = 0.405572 + 0.756937I
a = −0.41571 − 1.35816I
b = 0.62723 + 1.38475I
4.97233 − 2.95811I −1.13170 + 3.60082I
u = 0.405572 − 0.756937I
a = −0.41571 + 1.35816I
b = 0.62723 − 1.38475I
4.97233 + 2.95811I −1.13170 − 3.60082I
u = 1.189210 + 0.282581I
a = −1.088230 − 0.703914I
b = −0.228913 − 1.074910I
2.07423 − 1.22055I −3.51872 − 0.07112I
u = 1.189210 − 0.282581I
a = −1.088230 + 0.703914I
b = −0.228913 + 1.074910I
2.07423 + 1.22055I −3.51872 + 0.07112I
u = −0.889957 + 0.956699I
a = −0.521993 − 0.815508I
b = 0.302646 + 1.124860I
5.67221 + 1.09047I −3.82592 + 0.42258I
u = −0.889957 − 0.956699I
a = −0.521993 + 0.815508I
b = 0.302646 − 1.124860I
5.67221 − 1.09047I −3.82592 − 0.42258I
u = −1.023450 + 0.903197I
a = 0.541017 + 1.179680I
b = 0.695559 − 1.098830I
5.25155 + 5.76942I −4.89628 − 5.17142I
u = −1.023450 − 0.903197I
a = 0.541017 − 1.179680I
b = 0.695559 + 1.098830I
5.25155 − 5.76942I −4.89628 + 5.17142I
u = 0.509257 + 0.343539I
a = 0.44200 + 1.35055I
b = −0.332296 − 0.405177I
−0.575696 − 1.116820I −6.38496 + 6.15764I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.509257 − 0.343539I
a = 0.44200 − 1.35055I
b = −0.332296 + 0.405177I
−0.575696 + 1.116820I −6.38496 − 6.15764I
u = −0.550076 + 0.259421I
a = 1.50952 − 0.24668I
b = 0.988720 − 0.518259I
2.36168 + 3.34376I −0.22641 − 4.65236I
u = −0.550076 − 0.259421I
a = 1.50952 + 0.24668I
b = 0.988720 + 0.518259I
2.36168 − 3.34376I −0.22641 + 4.65236I
u = −0.841043 + 1.112380I
a = 0.821468 + 0.551752I
b = −1.23861 − 1.79456I
12.50880 − 2.04734I −0.610263 + 0.647242I
u = −0.841043 − 1.112380I
a = 0.821468 − 0.551752I
b = −1.23861 + 1.79456I
12.50880 + 2.04734I −0.610263 − 0.647242I
u = −1.13145 + 0.93287I
a = −0.73214 − 1.39000I
b = −1.52394 + 1.51302I
11.5470 + 9.4650I −1.80359 − 5.12935I
u = −1.13145 − 0.93287I
a = −0.73214 + 1.39000I
b = −1.52394 − 1.51302I
11.5470 − 9.4650I −1.80359 + 5.12935I
u = −0.441998
a = −1.82163
b = −0.952239
−1.60276 −5.18590
6
II. I
u
2
= hb − a, a
3
− a
2
+ 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
8
=
a
a
a
6
=
−a
2
− 1
−a
2
a
9
=
a
a
a
3
=
1
1
a
7
=
2a
a
a
10
=
−2a
2
+ a + 2
−a
2
+ a + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −a
2
+ 5a − 11
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u − 1)
3
c
2
, c
8
u
3
c
4
(u + 1)
3
c
5
, c
7
u
3
+ u
2
− 1
c
6
u
3
− u
2
+ 2u − 1
c
9
, c
10
u
3
+ u
2
+ 2u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
(y −1)
3
c
2
, c
8
y
3
c
5
, c
7
y
3
− y
2
+ 2y −1
c
6
, c
9
, c
10
y
3
+ 3y
2
+ 2y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.877439 + 0.744862I
b = 0.877439 + 0.744862I
1.37919 − 2.82812I −6.82789 + 2.41717I
u = 1.00000
a = 0.877439 − 0.744862I
b = 0.877439 − 0.744862I
1.37919 + 2.82812I −6.82789 − 2.41717I
u = 1.00000
a = −0.754878
b = −0.754878
−2.75839 −15.3440
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
3
)(u
18
− 4u
17
+ ··· + 3u − 1)
c
2
, c
8
u
3
(u
18
− u
17
+ ··· − 4u + 8)
c
3
((u − 1)
3
)(u
18
+ 4u
17
+ ··· + 11u + 1)
c
4
((u + 1)
3
)(u
18
− 4u
17
+ ··· + 3u − 1)
c
5
(u
3
+ u
2
− 1)(u
18
− 2u
17
+ ··· − 5u
2
+ 1)
c
6
(u
3
− u
2
+ 2u − 1)(u
18
− 2u
17
+ ··· + 2u − 1)
c
7
(u
3
+ u
2
− 1)(u
18
+ 2u
17
+ ··· + 18u − 17)
c
9
, c
10
(u
3
+ u
2
+ 2u + 1)(u
18
− 2u
17
+ ··· + 2u − 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
3
)(y
18
− 4y
17
+ ··· − 11y + 1)
c
2
, c
8
y
3
(y
18
− 21y
17
+ ··· − 592y + 64)
c
3
((y −1)
3
)(y
18
+ 24y
17
+ ··· − 11y + 1)
c
5
(y
3
− y
2
+ 2y −1)(y
18
+ 22y
17
+ ··· − 10y + 1)
c
6
, c
9
, c
10
(y
3
+ 3y
2
+ 2y −1)(y
18
+ 18y
17
+ ··· − 10y + 1)
c
7
(y
3
− y
2
+ 2y −1)(y
18
+ 10y
17
+ ··· − 1106y + 289)
12
