10
85
(K10a
86
)
A knot diagram
1
Linearized knot diagam
4 5 1 8 9 10 3 2 6 7
Solving Sequence
6,9
10 7
2,5
3 8 4 1
c
9
c
6
c
5
c
2
c
8
c
4
c
1
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h60294783u
28
− 88287773u
27
+ ··· + 34606354b − 88284553,
− 32141431u
28
+ 46155247u
27
+ ··· + 17303177a + 15723776, u
29
− 2u
28
+ ··· − 5u + 1i
I
u
2
= hb − 1, a + 1, u − 1i
* 2 irreducible components of dim
C
= 0, with total 30 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
= h6.03 × 10
7
u
28
− 8.83 × 10
7
u
27
+ · · · + 3.46 × 10
7
b − 8.83 × 10
7
, −3.21 ×
10
7
u
28
+4.62×10
7
u
27
+· · · +1.73×10
7
a+1.57×10
7
, u
29
−2u
28
+· · · − 5u +1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
7
=
u
−u
3
+ u
a
2
=
1.85755u
28
− 2.66744u
27
+ ··· + 14.0229u − 0.908722
−1.74230u
28
+ 2.55120u
27
+ ··· − 12.6457u + 2.55111
a
5
=
−u
u
a
3
=
1.82836u
28
− 2.55221u
27
+ ··· + 13.5669u − 0.794482
−1.71312u
28
+ 2.43597u
27
+ ··· − 12.1898u + 2.43687
a
8
=
3.97113u
28
− 4.76259u
27
+ ··· + 10.6810u − 4.64851
−4.16350u
28
+ 4.96257u
27
+ ··· − 16.5831u + 4.96513
a
4
=
−1.79416u
28
+ 2.57695u
27
+ ··· − 12.7088u + 1.57715
1.79416u
28
− 2.57695u
27
+ ··· + 12.7088u − 2.57715
a
1
=
−u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
289657216
17303177
u
28
−
414590110
17303177
u
27
+ ··· +
1326964710
17303177
u −
307662332
17303177
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
29
+ 2u
28
+ ··· − 9u + 1
c
2
u
29
− 5u
28
+ ··· + 2u + 2
c
4
u
29
+ 2u
28
+ ··· + u + 1
c
5
, c
6
, c
9
c
10
u
29
− 2u
28
+ ··· − 5u + 1
c
7
u
29
− 4u
27
+ ··· − 277u + 173
c
8
u
29
− 2u
28
+ ··· − 51u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
y
29
− 24y
28
+ ··· + 35y −1
c
2
y
29
+ 9y
28
+ ··· − 8y −4
c
4
y
29
+ 4y
28
+ ··· − y −1
c
5
, c
6
, c
9
c
10
y
29
− 36y
28
+ ··· − y −1
c
7
y
29
− 8y
28
+ ··· − 224637y −29929
c
8
y
29
− 36y
28
+ ··· + 2499y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.945899 + 0.527377I
a = −1.42859 − 0.46825I
b = 1.23966 − 0.80863I
5.88176 − 9.11420I 9.68617 + 7.34304I
u = −0.945899 − 0.527377I
a = −1.42859 + 0.46825I
b = 1.23966 + 0.80863I
5.88176 + 9.11420I 9.68617 − 7.34304I
u = 0.919400 + 0.642162I
a = −0.315900 + 0.817663I
b = 0.838796 − 0.254515I
5.23846 + 0.21078I 14.1588 + 0.1661I
u = 0.919400 − 0.642162I
a = −0.315900 − 0.817663I
b = 0.838796 + 0.254515I
5.23846 − 0.21078I 14.1588 − 0.1661I
u = −0.852692 + 0.102964I
a = −2.16375 + 1.13712I
b = 0.698160 − 0.148828I
4.90102 − 1.77997I 15.0441 + 4.1634I
u = −0.852692 − 0.102964I
a = −2.16375 − 1.13712I
b = 0.698160 + 0.148828I
4.90102 + 1.77997I 15.0441 − 4.1634I
u = −0.778126 + 0.317868I
a = 1.21478 + 0.99243I
b = −0.813260 + 0.652109I
1.06268 − 4.13663I 7.22942 + 7.89079I
u = −0.778126 − 0.317868I
a = 1.21478 − 0.99243I
b = −0.813260 − 0.652109I
1.06268 + 4.13663I 7.22942 − 7.89079I
u = 0.072018 + 0.813066I
a = −0.055406 − 0.155414I
b = −0.965881 − 0.630905I
2.76973 + 4.67347I 7.79912 − 5.64410I
u = 0.072018 − 0.813066I
a = −0.055406 + 0.155414I
b = −0.965881 + 0.630905I
2.76973 − 4.67347I 7.79912 + 5.64410I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.22499
a = −0.486025
b = −0.0200655
2.40006 1.29980
u = 0.696417
a = −3.92323
b = 3.47362
2.85586 −37.1850
u = 0.652302 + 0.186228I
a = 0.677568 − 0.199223I
b = −0.726039 − 0.441283I
1.260030 + 0.425153I 8.09157 − 0.86414I
u = 0.652302 − 0.186228I
a = 0.677568 + 0.199223I
b = −0.726039 + 0.441283I
1.260030 − 0.425153I 8.09157 + 0.86414I
u = −0.066518 + 0.465152I
a = 0.922681 + 0.795965I
b = 0.575911 + 0.420194I
−1.04879 + 1.39671I −0.58532 − 2.57538I
u = −0.066518 − 0.465152I
a = 0.922681 − 0.795965I
b = 0.575911 − 0.420194I
−1.04879 − 1.39671I −0.58532 + 2.57538I
u = −1.62115 + 0.04828I
a = −1.28657 + 0.61647I
b = 1.12014 − 1.06805I
9.19873 − 1.27855I 0
u = −1.62115 − 0.04828I
a = −1.28657 − 0.61647I
b = 1.12014 + 1.06805I
9.19873 + 1.27855I 0
u = −1.64256
a = 3.72388
b = −3.32176
11.1512 0
u = 1.65082 + 0.07616I
a = −1.56763 + 0.18293I
b = 0.985626 + 0.801694I
9.53565 + 5.57255I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.65082 − 0.07616I
a = −1.56763 − 0.18293I
b = 0.985626 − 0.801694I
9.53565 − 5.57255I 0
u = 1.67141 + 0.02331I
a = 1.76938 + 0.66852I
b = −0.801753 + 0.136783I
13.80520 + 2.24218I 0
u = 1.67141 − 0.02331I
a = 1.76938 − 0.66852I
b = −0.801753 − 0.136783I
13.80520 − 2.24218I 0
u = 1.69430 + 0.14926I
a = 2.01420 + 0.06571I
b = −1.47276 − 0.89148I
15.0062 + 11.7978I 0
u = 1.69430 − 0.14926I
a = 2.01420 − 0.06571I
b = −1.47276 + 0.89148I
15.0062 − 11.7978I 0
u = 0.175263 + 0.221780I
a = 3.00284 + 0.56107I
b = −0.809729 − 0.865243I
1.92999 + 0.70792I 4.69463 + 1.16490I
u = 0.175263 − 0.221780I
a = 3.00284 − 0.56107I
b = −0.809729 + 0.865243I
1.92999 − 0.70792I 4.69463 − 1.16490I
u = −1.71056 + 0.17143I
a = 1.059070 + 0.505719I
b = −0.934773 + 0.132892I
14.3721 − 3.4330I 0
u = −1.71056 − 0.17143I
a = 1.059070 − 0.505719I
b = −0.934773 − 0.132892I
14.3721 + 3.4330I 0
7
II. I
u
2
= hb − 1, a + 1, u − 1i
(i) Arc colorings
a
6
=
0
1
a
9
=
1
0
a
10
=
1
−1
a
7
=
1
0
a
2
=
−1
1
a
5
=
−1
1
a
3
=
−1
1
a
8
=
0
1
a
4
=
−1
2
a
1
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
6
u + 1
c
2
u
c
3
, c
7
, c
8
c
9
, c
10
u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
c
8
, c
9
, c
10
y −1
c
2
y
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = 1.00000
3.28987 12.0000
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)(u
29
+ 2u
28
+ ··· − 9u + 1)
c
2
u(u
29
− 5u
28
+ ··· + 2u + 2)
c
3
(u − 1)(u
29
+ 2u
28
+ ··· − 9u + 1)
c
4
(u + 1)(u
29
+ 2u
28
+ ··· + u + 1)
c
5
, c
6
(u + 1)(u
29
− 2u
28
+ ··· − 5u + 1)
c
7
(u − 1)(u
29
− 4u
27
+ ··· − 277u + 173)
c
8
(u − 1)(u
29
− 2u
28
+ ··· − 51u + 1)
c
9
, c
10
(u − 1)(u
29
− 2u
28
+ ··· − 5u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
(y −1)(y
29
− 24y
28
+ ··· + 35y −1)
c
2
y(y
29
+ 9y
28
+ ··· − 8y −4)
c
4
(y −1)(y
29
+ 4y
28
+ ··· − y −1)
c
5
, c
6
, c
9
c
10
(y −1)(y
29
− 36y
28
+ ··· − y −1)
c
7
(y −1)(y
29
− 8y
28
+ ··· − 224637y −29929)
c
8
(y −1)(y
29
− 36y
28
+ ··· + 2499y −1)
13
