10
147
(K10n
24
)
A knot diagram
1
Linearized knot diagam
4 9 6 8 1 3 9 5 6 3
Solving Sequence
6,9 4,10
3 7 1 2 5 8
c
9
c
3
c
6
c
10
c
2
c
5
c
8
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h713770382u
13
− 2219758738u
12
+ ··· + 3379396381b + 1752340181,
− 4583934274u
13
+ 15016790197u
12
+ ··· + 3379396381a + 53453066, u
14
− 3u
13
+ ··· + 6u + 1i
I
u
2
= h−u
3
− u
2
+ b − 4u − 1, 4u
3
+ 6u
2
+ a + 17u + 7, u
4
+ 2u
3
+ 5u
2
+ 4u + 1i
I
u
3
= hb, a − 1, u
3
+ u + 1i
* 3 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
= h7.14 × 10
8
u
13
− 2.22 × 10
9
u
12
+ · · · + 3.38 × 10
9
b + 1.75 × 10
9
, −4.58 ×
10
9
u
13
+1.50×10
10
u
12
+· · ·+3.38×10
9
a+5.35×10
7
, u
14
−3u
13
+· · ·+6u+1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
4
=
1.35644u
13
− 4.44363u
12
+ ··· + 29.0734u − 0.0158173
−0.211212u
13
+ 0.656851u
12
+ ··· − 7.55883u − 0.518536
a
10
=
1
u
2
a
3
=
1.35644u
13
− 4.44363u
12
+ ··· + 29.0734u − 0.0158173
−0.197062u
13
+ 0.657652u
12
+ ··· − 6.66932u − 0.144213
a
7
=
−0.664129u
13
+ 2.29385u
12
+ ··· − 12.7349u + 5.55469
0.0964891u
13
− 0.391604u
12
+ ··· + 1.77926u − 1.40855
a
1
=
−1.10709u
13
+ 3.17567u
12
+ ··· − 27.0789u − 7.56641
0.194643u
13
− 0.592457u
12
+ ··· + 4.85079u + 1.66551
a
2
=
1.55350u
13
− 5.10128u
12
+ ··· + 35.7427u + 0.128396
−0.197062u
13
+ 0.657652u
12
+ ··· − 6.66932u − 0.144213
a
5
=
0.144213u
13
− 0.629701u
12
+ ··· + 3.65663u − 5.80404
−0.0559147u
13
+ 0.229057u
12
+ ··· + 0.863012u + 1.51536
a
8
=
−0.760618u
13
+ 2.68545u
12
+ ··· − 14.5142u + 6.96324
0.0964891u
13
− 0.391604u
12
+ ··· + 1.77926u − 1.40855
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
4583624542
3379396381
u
13
+
13650120072
3379396381
u
12
+ ···−
173475920162
3379396381
u −
35870136224
3379396381
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
14
− 5u
13
+ ··· − 32u + 29
c
2
u
14
− u
13
+ ··· + 10u + 1
c
3
, c
6
u
14
+ u
13
+ ··· − 10u + 1
c
4
, c
8
u
14
− 2u
13
+ ··· − 3u + 2
c
5
u
14
+ u
13
+ ··· − 4u + 1
c
7
u
14
+ 8u
13
+ ··· − 19u + 4
c
9
u
14
− 3u
13
+ ··· + 6u + 1
c
10
u
14
+ 3u
13
+ ··· + 7u + 62
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
14
− 17y
13
+ ··· − 3924y + 841
c
2
y
14
+ 29y
13
+ ··· − 54y + 1
c
3
, c
6
y
14
+ 21y
13
+ ··· − 42y + 1
c
4
, c
8
y
14
− 8y
13
+ ··· + 19y + 4
c
5
y
14
+ y
13
+ ··· − 10y + 1
c
7
y
14
− 4y
13
+ ··· − 417y + 16
c
9
y
14
− 33y
13
+ ··· + 36y + 1
c
10
y
14
+ 25y
13
+ ··· + 20163y + 3844
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.237387 + 0.876423I
a = 0.004721 − 0.208169I
b = −0.869563 − 0.338885I
1.74438 + 2.05841I 4.16985 − 4.09365I
u = −0.237387 − 0.876423I
a = 0.004721 + 0.208169I
b = −0.869563 + 0.338885I
1.74438 − 2.05841I 4.16985 + 4.09365I
u = −0.595439 + 0.915402I
a = 0.915640 − 0.422475I
b = 1.027050 + 0.729987I
−1.83211 + 2.08733I −7.27574 − 2.80711I
u = −0.595439 − 0.915402I
a = 0.915640 + 0.422475I
b = 1.027050 − 0.729987I
−1.83211 − 2.08733I −7.27574 + 2.80711I
u = 0.021578 + 0.347833I
a = 1.92553 − 1.06606I
b = 0.029013 − 0.667088I
−0.11203 + 1.46789I −1.17938 − 4.69179I
u = 0.021578 − 0.347833I
a = 1.92553 + 1.06606I
b = 0.029013 + 0.667088I
−0.11203 − 1.46789I −1.17938 + 4.69179I
u = −0.113601 + 0.166050I
a = −1.16417 + 5.61112I
b = 0.064203 − 1.109710I
−3.57417 + 4.92202I −5.84899 − 5.58919I
u = −0.113601 − 0.166050I
a = −1.16417 − 5.61112I
b = 0.064203 + 1.109710I
−3.57417 − 4.92202I −5.84899 + 5.58919I
u = 2.25002 + 0.12421I
a = −0.037619 − 0.804931I
b = 0.43823 + 1.90805I
−13.29020 − 1.42119I −6.81603 + 0.70499I
u = 2.25002 − 0.12421I
a = −0.037619 + 0.804931I
b = 0.43823 − 1.90805I
−13.29020 + 1.42119I −6.81603 − 0.70499I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −2.43987 + 0.07821I
a = −0.036182 − 0.696454I
b = 0.47053 + 2.07372I
−8.54619 + 3.71322I −3.42485 − 2.09291I
u = −2.43987 − 0.07821I
a = −0.036182 + 0.696454I
b = 0.47053 − 2.07372I
−8.54619 − 3.71322I −3.42485 + 2.09291I
u = 2.61470 + 0.01753I
a = −0.107927 + 0.663193I
b = 0.34053 − 2.14014I
−12.2232 − 9.4176I −5.62486 + 4.99855I
u = 2.61470 − 0.01753I
a = −0.107927 − 0.663193I
b = 0.34053 + 2.14014I
−12.2232 + 9.4176I −5.62486 − 4.99855I
6
II.
I
u
2
= h−u
3
− u
2
+ b − 4u − 1, 4u
3
+ 6u
2
+ a + 17u + 7, u
4
+ 2u
3
+ 5u
2
+ 4u + 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
4
=
−4u
3
− 6u
2
− 17u − 7
u
3
+ u
2
+ 4u + 1
a
10
=
1
u
2
a
3
=
−4u
3
− 6u
2
− 17u − 7
−1
a
7
=
u
3
+ 2u
2
+ 5u + 4
−2u
3
− 3u
2
− 8u − 4
a
1
=
−4u
3
− 6u
2
− 17u − 7
u
3
+ 2u
2
+ 4u + 1
a
2
=
−4u
3
− 6u
2
− 17u − 6
−1
a
5
=
u
3
+ 2u
2
+ 5u + 4
−u
3
− 2u
2
− 5u − 3
a
8
=
3u
3
+ 5u
2
+ 13u + 8
−2u
3
− 3u
2
− 8u − 4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u
3
+ 12u
2
+ 32u + 12
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
4
− 4u
3
+ 5u
2
− 2u + 1
c
2
(u − 1)
4
c
3
, c
5
, c
6
(u
2
+ 1)
2
c
4
, c
8
, c
10
u
4
− u
2
+ 1
c
7
(u
2
− u + 1)
2
c
9
u
4
+ 2u
3
+ 5u
2
+ 4u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
4
− 6y
3
+ 11y
2
+ 6y + 1
c
2
(y −1)
4
c
3
, c
5
, c
6
(y + 1)
4
c
4
, c
8
, c
10
(y
2
− y + 1)
2
c
7
(y
2
+ y + 1)
2
c
9
y
4
+ 6y
3
+ 11y
2
− 6y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.133975I
a = 0.50000 − 1.86603I
b = −0.866025 + 0.500000I
− 2.02988I −2.00000 + 3.46410I
u = −0.500000 − 0.133975I
a = 0.50000 + 1.86603I
b = −0.866025 − 0.500000I
2.02988I −2.00000 − 3.46410I
u = −0.50000 + 1.86603I
a = 0.500000 − 0.133975I
b = 0.866025 + 0.500000I
2.02988I −2.00000 − 3.46410I
u = −0.50000 − 1.86603I
a = 0.500000 + 0.133975I
b = 0.866025 − 0.500000I
− 2.02988I −2.00000 + 3.46410I
10
III. I
u
3
= hb, a − 1, u
3
+ u + 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
4
=
1
0
a
10
=
1
u
2
a
3
=
1
u
2
a
7
=
u
−1
a
1
=
1
u
2
a
2
=
−u
2
+ 1
u
2
a
5
=
u
−1
a
8
=
u + 1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
+ 2u
2
+ u − 1
c
2
u
3
− 2u
2
+ u + 1
c
3
, c
5
, c
6
c
9
u
3
+ u + 1
c
4
, c
7
, c
8
(u + 1)
3
c
10
u
3
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
y
3
− 2y
2
+ 5y −1
c
3
, c
5
, c
6
c
9
y
3
+ 2y
2
+ y −1
c
4
, c
7
, c
8
(y −1)
3
c
10
y
3
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.341164 + 1.161540I
a = 1.00000
b = 0
−1.64493 −6.00000
u = 0.341164 − 1.161540I
a = 1.00000
b = 0
−1.64493 −6.00000
u = −0.682328
a = 1.00000
b = 0
−1.64493 −6.00000
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
+ 2u
2
+ u − 1)(u
4
− 4u
3
+ ··· − 2u + 1)(u
14
− 5u
13
+ ··· − 32u + 29)
c
2
((u − 1)
4
)(u
3
− 2u
2
+ u + 1)(u
14
− u
13
+ ··· + 10u + 1)
c
3
, c
6
((u
2
+ 1)
2
)(u
3
+ u + 1)(u
14
+ u
13
+ ··· − 10u + 1)
c
4
, c
8
((u + 1)
3
)(u
4
− u
2
+ 1)(u
14
− 2u
13
+ ··· − 3u + 2)
c
5
((u
2
+ 1)
2
)(u
3
+ u + 1)(u
14
+ u
13
+ ··· − 4u + 1)
c
7
((u + 1)
3
)(u
2
− u + 1)
2
(u
14
+ 8u
13
+ ··· − 19u + 4)
c
9
(u
3
+ u + 1)(u
4
+ 2u
3
+ ··· + 4u + 1)(u
14
− 3u
13
+ ··· + 6u + 1)
c
10
u
3
(u
4
− u
2
+ 1)(u
14
+ 3u
13
+ ··· + 7u + 62)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
3
− 2y
2
+ 5y −1)(y
4
− 6y
3
+ 11y
2
+ 6y + 1)
· (y
14
− 17y
13
+ ··· − 3924y + 841)
c
2
((y −1)
4
)(y
3
− 2y
2
+ 5y −1)(y
14
+ 29y
13
+ ··· − 54y + 1)
c
3
, c
6
((y + 1)
4
)(y
3
+ 2y
2
+ y −1)(y
14
+ 21y
13
+ ··· − 42y + 1)
c
4
, c
8
((y −1)
3
)(y
2
− y + 1)
2
(y
14
− 8y
13
+ ··· + 19y + 4)
c
5
((y + 1)
4
)(y
3
+ 2y
2
+ y −1)(y
14
+ y
13
+ ··· − 10y + 1)
c
7
((y −1)
3
)(y
2
+ y + 1)
2
(y
14
− 4y
13
+ ··· − 417y + 16)
c
9
(y
3
+ 2y
2
+ y −1)(y
4
+ 6y
3
+ ··· − 6y + 1)(y
14
− 33y
13
+ ··· + 36y + 1)
c
10
y
3
(y
2
− y + 1)
2
(y
14
+ 25y
13
+ ··· + 20163y + 3844)
16
