10
60
(K10a
1
)
A knot diagram
1
Linearized knot diagam
8 4 6 7 9 3 10 1 5 2
Solving Sequence
4,6
3 7
5,10
8 2 1 9
c
3
c
6
c
4
c
7
c
2
c
1
c
9
c
5
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
10
− u
9
+ 3u
8
− 2u
7
+ 3u
6
− 2u
5
− u
2
+ b, u
10
− u
9
+ 4u
8
− 3u
7
+ 6u
6
− 4u
5
+ 3u
4
− 2u
3
− u
2
+ a,
u
12
− u
11
+ 4u
10
− 3u
9
+ 7u
8
− 5u
7
+ 6u
6
− 4u
5
+ 2u
4
− 2u
3
+ u
2
+ 1i
I
u
2
= h−u
33
+ u
32
+ ··· + b + 1, −u
32
+ 2u
31
+ ··· + a − 2, u
34
− 2u
33
+ ··· − 3u + 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 50 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
10
− u
9
+ 3u
8
− 2u
7
+ 3u
6
− 2u
5
− u
2
+ b, u
10
− u
9
+ · · · − u
2
+
a, u
12
− u
11
+ · · · + u
2
+ 1i
(i) Arc colorings
a
4
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
7
=
u
u
3
+ u
a
5
=
u
4
+ u
2
+ 1
u
6
+ 2u
4
+ u
2
a
10
=
−u
10
+ u
9
− 4u
8
+ 3u
7
− 6u
6
+ 4u
5
− 3u
4
+ 2u
3
+ u
2
−u
10
+ u
9
− 3u
8
+ 2u
7
− 3u
6
+ 2u
5
+ u
2
a
8
=
u
11
− u
10
+ 3u
9
− 2u
8
+ 3u
7
− 2u
6
− 2u
3
−u
9
+ u
8
− 3u
7
+ 2u
6
− 3u
5
+ 2u
4
− u
3
a
2
=
u
2
+ 1
u
2
a
1
=
−u
10
+ u
9
− 4u
8
+ 3u
7
− 6u
6
+ 4u
5
− 4u
4
+ 2u
3
−u
10
+ u
9
− 3u
8
+ 2u
7
− 3u
6
+ 2u
5
− u
4
+ u
2
a
9
=
u
9
− u
8
+ 3u
7
− 2u
6
+ 4u
5
− 2u
4
+ 2u
3
u
11
− u
10
+ 4u
9
− 3u
8
+ 6u
7
− 4u
6
+ 4u
5
− 2u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
11
− 8u
10
+ 18u
9
− 26u
8
+ 34u
7
− 40u
6
+ 34u
5
− 24u
4
+ 16u
3
− 4u
2
+ 6u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
6
c
8
u
12
+ u
11
+ 4u
10
+ 3u
9
+ 7u
8
+ 5u
7
+ 6u
6
+ 4u
5
+ 2u
4
+ 2u
3
+ u
2
+ 1
c
2
, c
10
u
12
+ 7u
11
+ ··· + 2u + 1
c
4
, c
7
u
12
− u
11
+ ··· + 2u + 1
c
5
, c
9
u
12
− 5u
11
+ ··· − 12u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
8
y
12
+ 7y
11
+ ··· + 2y + 1
c
2
, c
10
y
12
− y
11
+ ··· + 6y + 1
c
4
, c
7
y
12
− 9y
11
+ ··· + 2y + 1
c
5
, c
9
y
12
+ 5y
11
+ ··· − 16y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.178968 + 0.877941I
a = −1.176440 − 0.426280I
b = −0.702552 + 0.572575I
−1.87720 − 1.89052I −4.24850 + 3.95054I
u = −0.178968 − 0.877941I
a = −1.176440 + 0.426280I
b = −0.702552 − 0.572575I
−1.87720 + 1.89052I −4.24850 − 3.95054I
u = 0.780097 + 0.281995I
a = 1.73075 + 0.13511I
b = 1.057890 + 0.528101I
−0.78013 − 3.73206I 3.21966 + 2.51013I
u = 0.780097 − 0.281995I
a = 1.73075 − 0.13511I
b = 1.057890 − 0.528101I
−0.78013 + 3.73206I 3.21966 − 2.51013I
u = −0.496677 + 1.117040I
a = −0.55633 + 2.12256I
b = 2.35694 + 1.72461I
−2.98532 − 7.52709I −1.88445 + 6.81034I
u = −0.496677 − 1.117040I
a = −0.55633 − 2.12256I
b = 2.35694 − 1.72461I
−2.98532 + 7.52709I −1.88445 − 6.81034I
u = 0.335900 + 1.207600I
a = −0.736004 − 0.940791I
b = 1.36295 − 1.08335I
−9.48086 + 3.21477I −6.88179 − 3.24710I
u = 0.335900 − 1.207600I
a = −0.736004 + 0.940791I
b = 1.36295 + 1.08335I
−9.48086 − 3.21477I −6.88179 + 3.24710I
u = 0.577185 + 1.164540I
a = 0.15978 − 1.92327I
b = 2.60480 − 1.08526I
−5.9276 + 13.9800I −2.44387 − 9.26853I
u = 0.577185 − 1.164540I
a = 0.15978 + 1.92327I
b = 2.60480 + 1.08526I
−5.9276 − 13.9800I −2.44387 + 9.26853I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.517537 + 0.434237I
a = 1.57824 − 0.67661I
b = 0.319971 − 0.990159I
1.31194 − 0.92364I 6.23895 + 2.73595I
u = −0.517537 − 0.434237I
a = 1.57824 + 0.67661I
b = 0.319971 + 0.990159I
1.31194 + 0.92364I 6.23895 − 2.73595I
6
II.
I
u
2
= h−u
33
+u
32
+· · ·+b+1, −u
32
+2u
31
+· · ·+a−2, u
34
−2u
33
+· · ·−3u+1i
(i) Arc colorings
a
4
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
7
=
u
u
3
+ u
a
5
=
u
4
+ u
2
+ 1
u
6
+ 2u
4
+ u
2
a
10
=
u
32
− 2u
31
+ ··· − 3u + 2
u
33
− u
32
+ ··· + 2u − 1
a
8
=
u
12
+ 3u
10
+ 5u
8
− 2u
7
+ 4u
6
− 4u
5
+ 2u
4
− 4u
3
+ u
2
+ 1
−u
33
+ u
32
+ ··· + 3u
2
− u
a
2
=
u
2
+ 1
u
2
a
1
=
3u
32
− 3u
31
+ ··· − 6u + 4
2u
33
+ 14u
31
+ ··· + 2u
3
+ u
2
a
9
=
2u
32
− 2u
31
+ ··· − 3u + 2
u
33
+ u
32
+ ··· + u
2
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −3u
33
+ 8u
32
− 27u
31
+ 61u
30
− 117u
29
+ 247u
28
− 346u
27
+
669u
26
− 778u
25
+ 1328u
24
− 1416u
23
+ 2034u
22
− 2108u
21
+ 2492u
20
− 2552u
19
+
2536u
18
− 2457u
17
+ 2183u
16
− 1857u
15
+ 1579u
14
− 1103u
13
+ 889u
12
− 533u
11
+
364u
10
− 219u
9
+ 98u
8
− 68u
7
+ 38u
6
− 24u
5
+ 20u
4
− 6u
3
+ 14u
2
− 7u + 7
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
6
c
8
u
34
+ 2u
33
+ ··· + 3u + 1
c
2
, c
10
u
34
+ 16u
33
+ ··· + u + 1
c
4
, c
7
u
34
− 2u
33
+ ··· − 183u + 73
c
5
, c
9
(u
17
+ 2u
16
+ ··· − 5u − 2)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
8
y
34
+ 16y
33
+ ··· + y + 1
c
2
, c
10
y
34
+ 4y
33
+ ··· + 17y + 1
c
4
, c
7
y
34
− 8y
33
+ ··· − 12903y + 5329
c
5
, c
9
(y
17
+ 10y
16
+ ··· − 23y −4)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.723313 + 0.731528I
a = 0.151866 + 0.654346I
b = 0.514055 − 0.693038I
−0.85292 − 6.04614I −0.59802 + 7.72564I
u = −0.723313 − 0.731528I
a = 0.151866 − 0.654346I
b = 0.514055 + 0.693038I
−0.85292 + 6.04614I −0.59802 − 7.72564I
u = −0.624264 + 0.668207I
a = 0.475559 − 0.697137I
b = −0.299501 − 0.231577I
1.18281 − 1.86595I 4.34837 + 4.33037I
u = −0.624264 − 0.668207I
a = 0.475559 + 0.697137I
b = −0.299501 + 0.231577I
1.18281 + 1.86595I 4.34837 − 4.33037I
u = −0.575012 + 0.946029I
a = 0.488103 − 0.422358I
b = 0.267905 − 0.921351I
0.36198 − 2.83643I 1.96538 + 0.68566I
u = −0.575012 − 0.946029I
a = 0.488103 + 0.422358I
b = 0.267905 + 0.921351I
0.36198 + 2.83643I 1.96538 − 0.68566I
u = 0.839419 + 0.294756I
a = −2.21863 − 0.02513I
b = −1.75177 − 0.94314I
−3.32961 − 8.73955I 0.19211 + 5.92158I
u = 0.839419 − 0.294756I
a = −2.21863 + 0.02513I
b = −1.75177 + 0.94314I
−3.32961 + 8.73955I 0.19211 − 5.92158I
u = −0.678441 + 0.881986I
a = −0.269083 − 0.051645I
b = −1.117340 − 0.103610I
−1.29776 + 0.72905I −2.79971 − 1.68011I
u = −0.678441 − 0.881986I
a = −0.269083 + 0.051645I
b = −1.117340 + 0.103610I
−1.29776 − 0.72905I −2.79971 + 1.68011I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.441434 + 1.051180I
a = −0.086124 − 0.253169I
b = −1.117340 − 0.103610I
−1.29776 + 0.72905I −2.79971 − 1.68011I
u = 0.441434 − 1.051180I
a = −0.086124 + 0.253169I
b = −1.117340 + 0.103610I
−1.29776 − 0.72905I −2.79971 + 1.68011I
u = −0.484889 + 1.050780I
a = 0.26211 − 1.43780I
b = −1.36154 − 1.18102I
−0.47242 − 3.20284I 2.38038 + 3.25895I
u = −0.484889 − 1.050780I
a = 0.26211 + 1.43780I
b = −1.36154 + 1.18102I
−0.47242 + 3.20284I 2.38038 − 3.25895I
u = −0.387508 + 1.102150I
a = 0.68089 + 1.93658I
b = 2.09444
−3.76357 −3.71974 + 0.I
u = −0.387508 − 1.102150I
a = 0.68089 − 1.93658I
b = 2.09444
−3.76357 −3.71974 + 0.I
u = 0.805751 + 0.171048I
a = −1.33086 + 0.63651I
b = −1.30277 + 0.63774I
−5.23887 − 0.57053I −2.63434 − 0.09683I
u = 0.805751 − 0.171048I
a = −1.33086 − 0.63651I
b = −1.30277 − 0.63774I
−5.23887 + 0.57053I −2.63434 + 0.09683I
u = 0.492477 + 1.076420I
a = 0.071402 + 0.579407I
b = 0.514055 + 0.693038I
−0.85292 + 6.04614I −0.59802 − 7.72564I
u = 0.492477 − 1.076420I
a = 0.071402 − 0.579407I
b = 0.514055 − 0.693038I
−0.85292 − 6.04614I −0.59802 + 7.72564I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.276836 + 1.167190I
a = 0.004108 + 1.012990I
b = −1.30277 + 0.63774I
−5.23887 − 0.57053I −2.63434 − 0.09683I
u = 0.276836 − 1.167190I
a = 0.004108 − 1.012990I
b = −1.30277 − 0.63774I
−5.23887 + 0.57053I −2.63434 + 0.09683I
u = 0.242359 + 1.211260I
a = 0.12569 − 1.56340I
b = 1.50375 − 0.40483I
−8.21063 − 5.43973I −5.49430 + 3.57628I
u = 0.242359 − 1.211260I
a = 0.12569 + 1.56340I
b = 1.50375 + 0.40483I
−8.21063 + 5.43973I −5.49430 − 3.57628I
u = 0.556877 + 1.148560I
a = −0.15813 + 1.53835I
b = −1.75177 + 0.94314I
−3.32961 + 8.73955I 0.19211 − 5.92158I
u = 0.556877 − 1.148560I
a = −0.15813 − 1.53835I
b = −1.75177 − 0.94314I
−3.32961 − 8.73955I 0.19211 + 5.92158I
u = 0.520828 + 1.178390I
a = 0.76467 − 1.29488I
b = 1.50375 + 0.40483I
−8.21063 + 5.43973I −5.49430 − 3.57628I
u = 0.520828 − 1.178390I
a = 0.76467 + 1.29488I
b = 1.50375 − 0.40483I
−8.21063 − 5.43973I −5.49430 + 3.57628I
u = 0.372098 + 0.537745I
a = −0.782608 − 0.762639I
b = 0.267905 + 0.921351I
0.36198 + 2.83643I 1.96538 − 0.68566I
u = 0.372098 − 0.537745I
a = −0.782608 + 0.762639I
b = 0.267905 − 0.921351I
0.36198 − 2.83643I 1.96538 + 0.68566I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.521356 + 0.372677I
a = 0.897739 + 0.802529I
b = −0.299501 − 0.231577I
1.18281 − 1.86595I 4.34837 + 4.33037I
u = 0.521356 − 0.372677I
a = 0.897739 − 0.802529I
b = −0.299501 + 0.231577I
1.18281 + 1.86595I 4.34837 − 4.33037I
u = −0.596010 + 0.210045I
a = −2.57670 + 0.72377I
b = −1.36154 + 1.18102I
−0.47242 + 3.20284I 2.38038 − 3.25895I
u = −0.596010 − 0.210045I
a = −2.57670 − 0.72377I
b = −1.36154 − 1.18102I
−0.47242 − 3.20284I 2.38038 + 3.25895I
13
III. I
u
3
= hb + u, a, u
2
+ u + 1i
(i) Arc colorings
a
4
=
1
0
a
6
=
0
u
a
3
=
1
−u − 1
a
7
=
u
u + 1
a
5
=
0
u
a
10
=
0
−u
a
8
=
u
u + 2
a
2
=
−u
−u − 1
a
1
=
1
−2u
a
9
=
0
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u + 4
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
6
, c
7
, c
10
u
2
− u + 1
c
3
, c
8
u
2
+ u + 1
c
5
, c
9
u
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
y
2
+ y + 1
c
5
, c
9
y
2
16
(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
17
IV. I
u
4
= hb + 1, a, u
2
+ u + 1i
(i) Arc colorings
a
4
=
1
0
a
6
=
0
u
a
3
=
1
−u − 1
a
7
=
u
u + 1
a
5
=
0
u
a
10
=
0
−1
a
8
=
u
2u + 1
a
2
=
−u
−u − 1
a
1
=
−u − 1
−2
a
9
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
6
, c
7
, c
10
u
2
− u + 1
c
3
, c
8
u
2
+ u + 1
c
5
, c
9
u
2
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
y
2
+ y + 1
c
5
, c
9
y
2
20
(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
21
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
2
− u + 1)
2
· (u
12
+ u
11
+ 4u
10
+ 3u
9
+ 7u
8
+ 5u
7
+ 6u
6
+ 4u
5
+ 2u
4
+ 2u
3
+ u
2
+ 1)
· (u
34
+ 2u
33
+ ··· + 3u + 1)
c
2
, c
10
((u
2
− u + 1)
2
)(u
12
+ 7u
11
+ ··· + 2u + 1)(u
34
+ 16u
33
+ ··· + u + 1)
c
3
, c
8
(u
2
+ u + 1)
2
· (u
12
+ u
11
+ 4u
10
+ 3u
9
+ 7u
8
+ 5u
7
+ 6u
6
+ 4u
5
+ 2u
4
+ 2u
3
+ u
2
+ 1)
· (u
34
+ 2u
33
+ ··· + 3u + 1)
c
4
, c
7
((u
2
− u + 1)
2
)(u
12
− u
11
+ ··· + 2u + 1)(u
34
− 2u
33
+ ··· − 183u + 73)
c
5
, c
9
u
4
(u
12
− 5u
11
+ ··· − 12u + 4)(u
17
+ 2u
16
+ ··· − 5u − 2)
2
22
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
8
((y
2
+ y + 1)
2
)(y
12
+ 7y
11
+ ··· + 2y + 1)(y
34
+ 16y
33
+ ··· + y + 1)
c
2
, c
10
((y
2
+ y + 1)
2
)(y
12
− y
11
+ ··· + 6y + 1)(y
34
+ 4y
33
+ ··· + 17y + 1)
c
4
, c
7
((y
2
+ y + 1)
2
)(y
12
− 9y
11
+ ··· + 2y + 1)
· (y
34
− 8y
33
+ ··· − 12903y + 5329)
c
5
, c
9
y
4
(y
12
+ 5y
11
+ ··· − 16y + 16)(y
17
+ 10y
16
+ ··· − 23y −4)
2
23
