10
146
(K10n
23
)
A knot diagram
1
Linearized knot diagam
7 5 7 9 2 4 1 6 3 2
Solving Sequence
2,7
1
4,8
3 6 5 10 9
c
1
c
7
c
3
c
6
c
5
c
10
c
9
c
2
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−469u
9
+ 1285u
8
+ ··· + 1534b + 422, −469u
9
+ 1285u
8
+ ··· + 3068a − 1879,
u
10
− 3u
9
+ 9u
8
− 16u
7
+ 24u
6
− 25u
5
+ 21u
4
− 4u
3
− 3u
2
+ 3u + 4i
I
u
2
= hb + 1, 3u
4
− 2u
3
+ a
2
+ 11u
2
− a − 5u + 7, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
I
u
3
= h−au + b + a + 1, a
2
− u, u
2
− u + 1i
* 3 irreducible components of dim
C
= 0, with total 24 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−469u
9
+ 1285u
8
+ · · · + 1534b + 422, −469u
9
+ 1285u
8
+ · · · +
3068a − 1879, u
10
− 3u
9
+ · · · + 3u + 4i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
1
=
1
u
2
a
4
=
0.152868u
9
− 0.418840u
8
+ ··· + 0.480769u + 0.612451
0.305737u
9
− 0.837679u
8
+ ··· − 1.03846u − 0.275098
a
8
=
u
u
3
+ u
a
3
=
0.152868u
9
− 0.418840u
8
+ ··· + 0.480769u + 0.612451
0.140808u
9
− 0.379400u
8
+ ··· − 0.307692u − 0.116037
a
6
=
0.0290091u
9
+ 0.0537810u
8
+ ··· − 0.711538u − 0.220665
0.0397653u
9
+ 0.0456323u
8
+ ··· + 0.153846u − 0.611473
a
5
=
0.0687744u
9
+ 0.0994133u
8
+ ··· − 0.557692u − 0.832138
0.0397653u
9
+ 0.0456323u
8
+ ··· + 0.153846u − 0.611473
a
10
=
u
2
+ 1
u
2
a
9
=
−0.0602999u
9
+ 0.197197u
8
+ ··· + 1.05769u + 0.857562
0.0162973u
9
+ 0.108866u
8
+ ··· + 1.03846u + 0.241199
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
171
767
u
9
−
269
767
u
8
+
1052
767
u
7
−
1818
767
u
6
+
2398
767
u
5
−
4011
767
u
4
+
2685
767
u
3
−
3379
767
u
2
+
37
13
u −
2378
767
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
10
− 3u
9
+ 9u
8
− 16u
7
+ 24u
6
− 25u
5
+ 21u
4
− 4u
3
− 3u
2
+ 3u + 4
c
2
, c
3
, c
5
c
6
u
10
+ u
8
+ u
7
+ 5u
6
+ 2u
4
+ 3u
3
+ 2u
2
+ 1
c
4
u
10
− 3u
9
+ 3u
8
+ 2u
7
− 6u
6
+ 3u
5
+ 3u
4
− 4u
3
+ 3u
2
− 3u + 2
c
8
, c
9
u
10
+ 2u
9
+ 9u
8
+ 7u
7
+ 30u
6
+ 6u
5
+ 41u
4
+ 22u
2
+ 4
c
10
u
10
+ 9u
9
+ ··· − 33u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
10
+ 9y
9
+ ··· − 33y + 16
c
2
, c
3
, c
5
c
6
y
10
+ 2y
9
+ 11y
8
+ 13y
7
+ 33y
6
+ 20y
5
+ 26y
4
+ 9y
3
+ 8y
2
+ 4y + 1
c
4
y
10
− 3y
9
+ 9y
8
− 16y
7
+ 24y
6
− 25y
5
+ 21y
4
− 4y
3
− 3y
2
+ 3y + 4
c
8
, c
9
y
10
+ 14y
9
+ ··· + 176y + 16
c
10
y
10
− 15y
9
+ ··· + 5343y + 256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.741866 + 0.796341I
a = 0.500393 + 0.239842I
b = −0.625089 + 0.778917I
1.60483 + 1.51336I 1.256588 − 0.171947I
u = 0.741866 − 0.796341I
a = 0.500393 − 0.239842I
b = −0.625089 − 0.778917I
1.60483 − 1.51336I 1.256588 + 0.171947I
u = 1.077560 + 0.740596I
a = 0.030843 − 0.749210I
b = 0.94514 − 1.33248I
1.74604 + 4.90489I 2.53483 − 7.39457I
u = 1.077560 − 0.740596I
a = 0.030843 + 0.749210I
b = 0.94514 + 1.33248I
1.74604 − 4.90489I 2.53483 + 7.39457I
u = −0.429682 + 0.277960I
a = 0.69620 + 1.42291I
b = 0.722559 + 0.567039I
−1.23090 − 1.07704I −4.33290 + 2.58024I
u = −0.429682 − 0.277960I
a = 0.69620 − 1.42291I
b = 0.722559 − 0.567039I
−1.23090 + 1.07704I −4.33290 − 2.58024I
u = −0.25937 + 1.52583I
a = −0.571923 + 0.727637I
b = 1.66770 + 0.84950I
−7.19127 − 3.97850I −1.38540 + 2.06163I
u = −0.25937 − 1.52583I
a = −0.571923 − 0.727637I
b = 1.66770 − 0.84950I
−7.19127 + 3.97850I −1.38540 − 2.06163I
u = 0.36963 + 1.73551I
a = −0.530514 − 0.624791I
b = 1.78968 − 0.93001I
−6.44324 + 10.56100I −0.07312 − 6.56398I
u = 0.36963 − 1.73551I
a = −0.530514 + 0.624791I
b = 1.78968 + 0.93001I
−6.44324 − 10.56100I −0.07312 + 6.56398I
5
II.
I
u
2
= hb + 1, 3u
4
− 2u
3
+ a
2
+ 11u
2
− a − 5u + 7, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
1
=
1
u
2
a
4
=
a
−1
a
8
=
u
u
3
+ u
a
3
=
a
−u
2
a − 1
a
6
=
−u
4
+ u
3
+ au − 4u
2
+ 2u − 3
−au + u
a
5
=
−u
4
+ u
3
− 4u
2
+ 3u − 3
−au + u
a
10
=
u
2
+ 1
u
2
a
9
=
−u
4
a + u
4
− 3u
2
a + 4u
2
− a + 3
u
4
a − u
3
a + 3u
2
a + u
3
− 2au + a + 2u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
4
− 4u
3
+ 16u
2
− 12u + 10
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)
2
c
2
, c
3
, c
5
c
6
u
10
− u
9
+ 2u
8
− 2u
7
+ 4u
6
− 4u
5
+ 9u
4
− 7u
3
+ 8u
2
− 4u + 1
c
4
(u
5
+ u
4
− u
2
+ u + 1)
2
c
8
, c
9
u
10
+ u
9
+ 2u
8
− 2u
7
+ 6u
6
+ 10u
5
+ 11u
4
+ 27u
3
+ 6u
2
+ 10u + 29
c
10
(u
5
+ 7u
4
+ 16u
3
+ 13u
2
+ 3u − 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)
2
c
2
, c
3
, c
5
c
6
y
10
+ 3y
9
+ 8y
8
+ 22y
7
+ 38y
6
+ 54y
5
+ 77y
4
+ 71y
3
+ 26y
2
+ 1
c
4
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)
2
c
8
, c
9
y
10
+ 3y
9
+ ··· + 248y + 841
c
10
(y
5
− 17y
4
+ 80y
3
− 59y
2
+ 35y −1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.233677 + 0.885557I
a = 1.186080 + 0.428672I
b = −1.00000
1.47006 + 2.21397I −0.88568 − 4.22289I
u = 0.233677 + 0.885557I
a = −0.186079 − 0.428672I
b = −1.00000
1.47006 + 2.21397I −0.88568 − 4.22289I
u = 0.233677 − 0.885557I
a = 1.186080 − 0.428672I
b = −1.00000
1.47006 − 2.21397I −0.88568 + 4.22289I
u = 0.233677 − 0.885557I
a = −0.186079 + 0.428672I
b = −1.00000
1.47006 − 2.21397I −0.88568 + 4.22289I
u = 0.416284
a = 0.50000 + 2.55355I
b = −1.00000
4.17205 7.60880
u = 0.416284
a = 0.50000 − 2.55355I
b = −1.00000
4.17205 7.60880
u = 0.05818 + 1.69128I
a = 0.518923 + 0.634033I
b = −1.00000
−7.66842 + 3.33174I −1.91874 − 2.36228I
u = 0.05818 + 1.69128I
a = 0.481077 − 0.634033I
b = −1.00000
−7.66842 + 3.33174I −1.91874 − 2.36228I
u = 0.05818 − 1.69128I
a = 0.518923 − 0.634033I
b = −1.00000
−7.66842 − 3.33174I −1.91874 + 2.36228I
u = 0.05818 − 1.69128I
a = 0.481077 + 0.634033I
b = −1.00000
−7.66842 − 3.33174I −1.91874 + 2.36228I
9
III. I
u
3
= h−au + b + a + 1, a
2
− u, u
2
− u + 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
1
=
1
u − 1
a
4
=
a
au − a − 1
a
8
=
u
u − 1
a
3
=
a
−1
a
6
=
u − 1
−au
a
5
=
−au + u − 1
−au
a
10
=
u
u − 1
a
9
=
−au + u + 1
au − a + 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u + 8
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
(u
2
− u + 1)
2
c
2
, c
3
, c
5
c
6
(u
2
+ 1)
2
c
4
u
4
− u
2
+ 1
c
7
(u
2
+ u + 1)
2
c
8
u
4
− 2u
3
+ 2u
2
− 4u + 4
c
9
u
4
+ 2u
3
+ 2u
2
+ 4u + 4
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
10
(y
2
+ y + 1)
2
c
2
, c
3
, c
5
c
6
(y + 1)
4
c
4
(y
2
− y + 1)
2
c
8
, c
9
y
4
− 4y
2
+ 16
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.866025 + 0.500000I
b = −1.86603 + 0.50000I
3.28987 + 2.02988I 6.00000 − 3.46410I
u = 0.500000 + 0.866025I
a = −0.866025 − 0.500000I
b = −0.133975 − 0.500000I
3.28987 + 2.02988I 6.00000 − 3.46410I
u = 0.500000 − 0.866025I
a = 0.866025 − 0.500000I
b = −1.86603 − 0.50000I
3.28987 − 2.02988I 6.00000 + 3.46410I
u = 0.500000 − 0.866025I
a = −0.866025 + 0.500000I
b = −0.133975 + 0.500000I
3.28987 − 2.02988I 6.00000 + 3.46410I
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)
2
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)
2
· (u
10
− 3u
9
+ 9u
8
− 16u
7
+ 24u
6
− 25u
5
+ 21u
4
− 4u
3
− 3u
2
+ 3u + 4)
c
2
, c
3
, c
5
c
6
(u
2
+ 1)
2
(u
10
+ u
8
+ u
7
+ 5u
6
+ 2u
4
+ 3u
3
+ 2u
2
+ 1)
· (u
10
− u
9
+ 2u
8
− 2u
7
+ 4u
6
− 4u
5
+ 9u
4
− 7u
3
+ 8u
2
− 4u + 1)
c
4
(u
4
− u
2
+ 1)(u
5
+ u
4
− u
2
+ u + 1)
2
· (u
10
− 3u
9
+ 3u
8
+ 2u
7
− 6u
6
+ 3u
5
+ 3u
4
− 4u
3
+ 3u
2
− 3u + 2)
c
7
(u
2
+ u + 1)
2
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)
2
· (u
10
− 3u
9
+ 9u
8
− 16u
7
+ 24u
6
− 25u
5
+ 21u
4
− 4u
3
− 3u
2
+ 3u + 4)
c
8
(u
4
− 2u
3
+ 2u
2
− 4u + 4)
· (u
10
+ u
9
+ 2u
8
− 2u
7
+ 6u
6
+ 10u
5
+ 11u
4
+ 27u
3
+ 6u
2
+ 10u + 29)
· (u
10
+ 2u
9
+ 9u
8
+ 7u
7
+ 30u
6
+ 6u
5
+ 41u
4
+ 22u
2
+ 4)
c
9
(u
4
+ 2u
3
+ 2u
2
+ 4u + 4)
· (u
10
+ u
9
+ 2u
8
− 2u
7
+ 6u
6
+ 10u
5
+ 11u
4
+ 27u
3
+ 6u
2
+ 10u + 29)
· (u
10
+ 2u
9
+ 9u
8
+ 7u
7
+ 30u
6
+ 6u
5
+ 41u
4
+ 22u
2
+ 4)
c
10
(u
2
− u + 1)
2
(u
5
+ 7u
4
+ 16u
3
+ 13u
2
+ 3u − 1)
2
· (u
10
+ 9u
9
+ ··· − 33u + 16)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
2
+ y + 1)
2
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)
2
· (y
10
+ 9y
9
+ ··· − 33y + 16)
c
2
, c
3
, c
5
c
6
(y + 1)
4
· (y
10
+ 2y
9
+ 11y
8
+ 13y
7
+ 33y
6
+ 20y
5
+ 26y
4
+ 9y
3
+ 8y
2
+ 4y + 1)
· (y
10
+ 3y
9
+ 8y
8
+ 22y
7
+ 38y
6
+ 54y
5
+ 77y
4
+ 71y
3
+ 26y
2
+ 1)
c
4
(y
2
− y + 1)
2
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)
2
· (y
10
− 3y
9
+ 9y
8
− 16y
7
+ 24y
6
− 25y
5
+ 21y
4
− 4y
3
− 3y
2
+ 3y + 4)
c
8
, c
9
(y
4
− 4y
2
+ 16)(y
10
+ 3y
9
+ ··· + 248y + 841)
· (y
10
+ 14y
9
+ ··· + 176y + 16)
c
10
(y
2
+ y + 1)
2
(y
5
− 17y
4
+ 80y
3
− 59y
2
+ 35y −1)
2
· (y
10
− 15y
9
+ ··· + 5343y + 256)
15
