10
50
(K10a
82
)
A knot diagram
1
Linearized knot diagam
6 10 7 8 9 1 5 4 2 3
Solving Sequence
5,7
8 4 9
1,3
6 2 10
c
7
c
4
c
8
c
3
c
6
c
1
c
10
c
2
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
28
− 2u
27
+ ··· + b + 1, 2u
28
− 2u
27
+ ··· + a + 2, u
29
− 2u
28
+ ··· + u − 1i
I
u
2
= hb, u
2
+ a + 1, u
3
+ u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 32 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
28
−2u
27
+· · ·+b +1, 2u
28
−2u
27
+· · ·+a +2, u
29
−2u
28
+· · ·+u −1i
(i) Arc colorings
a
5
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
4
=
u
u
3
+ u
a
9
=
u
2
+ 1
u
4
+ 2u
2
a
1
=
−2u
28
+ 2u
27
+ ··· − 6u − 2
−u
28
+ 2u
27
+ ··· + u − 1
a
3
=
u
3
+ 2u
u
3
+ u
a
6
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
2
=
−u
16
− 7u
14
+ ··· − 6u − 1
u
28
− 2u
27
+ ··· − u
2
+ 1
a
10
=
−u
28
+ u
27
+ ··· − 5u − 1
u
17
+ 7u
15
+ ··· + 6u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
28
−2u
27
+ 17u
26
−28u
25
+ 121u
24
−169u
23
+ 476u
22
−572u
21
+
1124u
20
−1170u
19
+ 1569u
18
−1418u
17
+ 1069u
16
−834u
15
−98u
14
+ 112u
13
−636u
12
+
544u
11
− 270u
10
+ 426u
9
− 12u
8
+ 183u
7
− 90u
6
− 34u
5
− 38u
4
− 76u
3
+ 29u
2
+ 2u − 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
29
+ u
28
+ ··· − 4u − 8
c
2
, c
9
, c
10
u
29
− 4u
28
+ ··· + 2u − 1
c
3
, c
5
u
29
+ 2u
28
+ ··· − 15u − 9
c
4
, c
7
, c
8
u
29
− 2u
28
+ ··· + u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
29
+ 21y
28
+ ··· + 144y −64
c
2
, c
9
, c
10
y
29
− 30y
28
+ ··· + 18y −1
c
3
, c
5
y
29
− 24y
28
+ ··· + 621y −81
c
4
, c
7
, c
8
y
29
+ 24y
28
+ ··· + 13y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.872970 + 0.113870I
a = 0.23209 + 2.29662I
b = −0.56484 + 1.49174I
−12.27840 − 6.66801I −13.30046 + 3.89200I
u = 0.872970 − 0.113870I
a = 0.23209 − 2.29662I
b = −0.56484 − 1.49174I
−12.27840 + 6.66801I −13.30046 − 3.89200I
u = −0.824312
a = −1.01744
b = −1.30242
−7.43008 −12.5870
u = 0.814174 + 0.046599I
a = −0.17300 − 2.55555I
b = 0.215027 − 1.248980I
−5.26114 − 2.70743I −11.83350 + 3.32702I
u = 0.814174 − 0.046599I
a = −0.17300 + 2.55555I
b = 0.215027 + 1.248980I
−5.26114 + 2.70743I −11.83350 − 3.32702I
u = 0.050561 + 1.224810I
a = 1.179120 − 0.735696I
b = −0.603790 − 0.612719I
1.43725 − 1.10103I −6.03106 − 0.28755I
u = 0.050561 − 1.224810I
a = 1.179120 + 0.735696I
b = −0.603790 + 0.612719I
1.43725 + 1.10103I −6.03106 + 0.28755I
u = 0.438893 + 1.153290I
a = 0.614951 + 0.762748I
b = 0.45548 + 1.52023I
−9.09072 + 1.97634I −10.56391 − 0.15391I
u = 0.438893 − 1.153290I
a = 0.614951 − 0.762748I
b = 0.45548 − 1.52023I
−9.09072 − 1.97634I −10.56391 + 0.15391I
u = −0.566873 + 0.506506I
a = −0.914731 + 0.813821I
b = 0.112616 + 1.303260I
−6.34917 + 2.02688I −11.64196 − 3.46616I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.566873 − 0.506506I
a = −0.914731 − 0.813821I
b = 0.112616 − 1.303260I
−6.34917 − 2.02688I −11.64196 + 3.46616I
u = 0.357598 + 1.229040I
a = −0.75023 − 1.33832I
b = −0.063501 − 1.233240I
−1.62082 − 1.51334I −8.49380 + 0.41799I
u = 0.357598 − 1.229040I
a = −0.75023 + 1.33832I
b = −0.063501 + 1.233240I
−1.62082 + 1.51334I −8.49380 − 0.41799I
u = −0.255230 + 1.288030I
a = 0.111769 − 0.476267I
b = −0.607413 + 0.112242I
2.53302 + 3.25312I −0.46847 − 3.58405I
u = −0.255230 − 1.288030I
a = 0.111769 + 0.476267I
b = −0.607413 − 0.112242I
2.53302 − 3.25312I −0.46847 + 3.58405I
u = −0.075468 + 1.316000I
a = −0.969152 + 0.088875I
b = 0.538894 + 0.689414I
4.46963 + 2.10537I −0.57633 − 3.98592I
u = −0.075468 − 1.316000I
a = −0.969152 − 0.088875I
b = 0.538894 − 0.689414I
4.46963 − 2.10537I −0.57633 + 3.98592I
u = −0.369778 + 1.269420I
a = −0.182052 + 0.874747I
b = 1.298970 − 0.143296I
−3.48935 + 4.29283I −8.53955 − 3.19264I
u = −0.369778 − 1.269420I
a = −0.182052 − 0.874747I
b = 1.298970 + 0.143296I
−3.48935 − 4.29283I −8.53955 + 3.19264I
u = 0.361886 + 1.302780I
a = 1.27373 + 1.39712I
b = −0.338315 + 1.255880I
−1.04610 − 6.94187I −7.09973 + 6.05967I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.361886 − 1.302780I
a = 1.27373 − 1.39712I
b = −0.338315 − 1.255880I
−1.04610 + 6.94187I −7.09973 − 6.05967I
u = −0.645651
a = 0.563691
b = 0.525371
−1.50367 −5.88400
u = 0.389029 + 1.350370I
a = −1.50604 − 1.16997I
b = 0.63881 − 1.44580I
−7.67865 − 11.19890I −9.19156 + 6.17598I
u = 0.389029 − 1.350370I
a = −1.50604 + 1.16997I
b = 0.63881 + 1.44580I
−7.67865 + 11.19890I −9.19156 − 6.17598I
u = −0.14677 + 1.42338I
a = 0.845011 + 0.480671I
b = −0.257766 − 1.113060I
−0.14603 + 4.37313I −7.64888 − 4.01970I
u = −0.14677 − 1.42338I
a = 0.845011 − 0.480671I
b = −0.257766 + 1.113060I
−0.14603 − 4.37313I −7.64888 + 4.01970I
u = −0.274649 + 0.285133I
a = 0.844421 − 1.049180I
b = −0.175226 − 0.644435I
−0.389560 + 0.938777I −6.80996 − 7.32576I
u = −0.274649 − 0.285133I
a = 0.844421 + 1.049180I
b = −0.175226 + 0.644435I
−0.389560 − 0.938777I −6.80996 + 7.32576I
u = 0.277276
a = −2.75803
b = 0.479164
−2.07267 −2.13090
7
II. I
u
2
= hb, u
2
+ a + 1, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
5
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
4
=
u
−u
2
− u − 1
a
9
=
u
2
+ 1
u
2
+ u + 1
a
1
=
−u
2
− 1
0
a
3
=
−u
2
− 1
−u
2
− u − 1
a
6
=
1
0
a
2
=
−u
2
− 1
0
a
10
=
0
u
2
+ u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5u
2
− 4u − 16
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
3
c
2
(u + 1)
3
c
3
, c
5
u
3
+ u
2
− 1
c
4
u
3
− u
2
+ 2u − 1
c
7
, c
8
u
3
+ u
2
+ 2u + 1
c
9
, c
10
(u − 1)
3
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
3
c
2
, c
9
, c
10
(y −1)
3
c
3
, c
5
y
3
− y
2
+ 2y −1
c
4
, c
7
, c
8
y
3
+ 3y
2
+ 2y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = 0.662359 + 0.562280I
b = 0
1.37919 + 2.82812I −6.82789 − 2.41717I
u = −0.215080 − 1.307140I
a = 0.662359 − 0.562280I
b = 0
1.37919 − 2.82812I −6.82789 + 2.41717I
u = −0.569840
a = −1.32472
b = 0
−2.75839 −15.3440
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
u
3
(u
29
+ u
28
+ ··· − 4u − 8)
c
2
((u + 1)
3
)(u
29
− 4u
28
+ ··· + 2u − 1)
c
3
, c
5
(u
3
+ u
2
− 1)(u
29
+ 2u
28
+ ··· − 15u − 9)
c
4
(u
3
− u
2
+ 2u − 1)(u
29
− 2u
28
+ ··· + u − 1)
c
7
, c
8
(u
3
+ u
2
+ 2u + 1)(u
29
− 2u
28
+ ··· + u − 1)
c
9
, c
10
((u − 1)
3
)(u
29
− 4u
28
+ ··· + 2u − 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
3
(y
29
+ 21y
28
+ ··· + 144y −64)
c
2
, c
9
, c
10
((y −1)
3
)(y
29
− 30y
28
+ ··· + 18y −1)
c
3
, c
5
(y
3
− y
2
+ 2y −1)(y
29
− 24y
28
+ ··· + 621y −81)
c
4
, c
7
, c
8
(y
3
+ 3y
2
+ 2y −1)(y
29
+ 24y
28
+ ··· + 13y −1)
13
