10
82
(K10a
83
)
A knot diagram
1
Linearized knot diagam
7 8 10 1 2 9 5 6 3 4
Solving Sequence
1,5 4,8
7 2 6 10 3 9
c
4
c
7
c
1
c
5
c
10
c
3
c
9
c
2
, c
6
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h1790814371u
31
+ 3053908485u
30
+ ··· + 15215838414b + 1796669401,
9786061617u
31
+ 13386015963u
30
+ ··· + 5071946138a + 29865915991, u
32
+ 2u
31
+ ··· − u + 1i
I
u
2
= hb, a + 1, u + 1i
* 2 irreducible components of dim
C
= 0, with total 33 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
= h1.79 × 10
9
u
31
+ 3.05 × 10
9
u
30
+ · · · + 1.52 × 10
10
b + 1.80 × 10
9
, 9.79 ×
10
9
u
31
+1.34×10
10
u
30
+· · ·+5.07×10
9
a+2.99×10
10
, u
32
+2u
31
+· · ·−u+1i
(i) Arc colorings
a
1
=
0
u
a
5
=
1
0
a
4
=
1
−u
2
a
8
=
−1.92945u
31
− 2.63923u
30
+ ··· + 9.12108u − 5.88845
−0.117694u
31
− 0.200706u
30
+ ··· + 2.07712u − 0.118079
a
7
=
−2.04714u
31
− 2.83993u
30
+ ··· + 11.1982u − 6.00653
−0.117694u
31
− 0.200706u
30
+ ··· + 2.07712u − 0.118079
a
2
=
−2.22835u
31
− 3.03012u
30
+ ··· + 0.715268u + 0.285838
−0.998715u
31
− 0.999372u
30
+ ··· + 1.28281u − 0.999382
a
6
=
2.22354u
31
+ 3.04014u
30
+ ··· − 10.0154u + 6.22362
−u
5
+ 3u
3
− u
a
10
=
u
−u
3
+ u
a
3
=
−u
2
+ 1
u
4
− 2u
2
a
9
=
−u
3
+ 2u
u
5
− 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
18210579048
2535973069
u
31
+
32359838926
2535973069
u
30
+ ··· −
94883406442
2535973069
u +
41946667180
2535973069
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
32
+ 2u
31
+ ··· + 12u + 8
c
2
u
32
+ 11u
30
+ ··· + 13u − 1
c
3
, c
4
, c
9
c
10
u
32
+ 2u
31
+ ··· − u + 1
c
5
u
32
− 2u
31
+ ··· + u − 1
c
6
, c
8
u
32
+ 2u
31
+ ··· − 13u − 1
c
7
u
32
− 5u
31
+ ··· − 6u + 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
32
+ 30y
31
+ ··· + 240y + 64
c
2
y
32
+ 22y
31
+ ··· − 121y + 1
c
3
, c
4
, c
9
c
10
y
32
− 38y
31
+ ··· − 5y + 1
c
5
y
32
− 6y
31
+ ··· − 5y + 1
c
6
, c
8
y
32
− 18y
31
+ ··· − 81y + 1
c
7
y
32
− 9y
31
+ ··· − 32y + 4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.820983 + 0.567595I
a = −1.27469 + 0.62091I
b = 1.088800 + 0.850114I
0.60537 − 9.61260I −2.87987 + 8.20248I
u = 0.820983 − 0.567595I
a = −1.27469 − 0.62091I
b = 1.088800 − 0.850114I
0.60537 + 9.61260I −2.87987 − 8.20248I
u = 0.795955 + 0.349102I
a = 1.45784 − 0.39446I
b = −1.136450 − 0.835713I
−2.75563 − 4.13382I −6.93448 + 6.73749I
u = 0.795955 − 0.349102I
a = 1.45784 + 0.39446I
b = −1.136450 + 0.835713I
−2.75563 + 4.13382I −6.93448 − 6.73749I
u = −0.643643 + 0.579820I
a = 0.109445 + 0.730653I
b = −0.758624 + 0.110290I
−1.27204 + 1.92248I −7.80216 − 5.91516I
u = −0.643643 − 0.579820I
a = 0.109445 − 0.730653I
b = −0.758624 − 0.110290I
−1.27204 − 1.92248I −7.80216 + 5.91516I
u = −1.076160 + 0.444148I
a = −0.311615 − 0.602654I
b = 0.691368 + 0.318391I
−0.800175 − 0.941991I −6.40540 + 5.25085I
u = −1.076160 − 0.444148I
a = −0.311615 + 0.602654I
b = 0.691368 − 0.318391I
−0.800175 + 0.941991I −6.40540 − 5.25085I
u = −0.788048
a = −0.997928
b = 0.333761
−1.36694 −7.37900
u = 0.102445 + 0.771273I
a = 0.249085 − 0.151496I
b = 0.853465 − 0.688304I
2.78249 + 5.16401I 0.17525 − 5.43243I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.102445 − 0.771273I
a = 0.249085 + 0.151496I
b = 0.853465 + 0.688304I
2.78249 − 5.16401I 0.17525 + 5.43243I
u = 0.560858 + 0.310184I
a = −0.155519 + 0.637386I
b = 0.671965 − 1.149150I
1.86601 − 2.61443I 0.82365 + 8.13996I
u = 0.560858 − 0.310184I
a = −0.155519 − 0.637386I
b = 0.671965 + 1.149150I
1.86601 + 2.61443I 0.82365 − 8.13996I
u = −0.598750 + 0.114970I
a = 0.25826 − 3.79474I
b = −0.135421 − 0.360183I
0.576409 + 0.313871I 8.1378 + 17.1065I
u = −0.598750 − 0.114970I
a = 0.25826 + 3.79474I
b = −0.135421 + 0.360183I
0.576409 − 0.313871I 8.1378 − 17.1065I
u = −0.086458 + 0.449548I
a = −0.783456 + 0.459529I
b = −0.610958 + 0.536174I
−0.227616 + 1.394370I −2.60146 − 4.04487I
u = −0.086458 − 0.449548I
a = −0.783456 − 0.459529I
b = −0.610958 − 0.536174I
−0.227616 − 1.394370I −2.60146 + 4.04487I
u = −1.55208
a = −2.62954
b = 1.76871
−3.73390 0
u = −1.57850 + 0.06009I
a = −0.52697 − 1.39477I
b = 0.56830 + 1.70360I
−5.46664 + 3.81790I 0
u = −1.57850 − 0.06009I
a = −0.52697 + 1.39477I
b = 0.56830 − 1.70360I
−5.46664 − 3.81790I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.60015 + 0.02565I
a = 0.830347 + 1.077280I
b = −0.363802 + 0.595725I
−7.09081 − 0.79638I 0
u = 1.60015 − 0.02565I
a = 0.830347 − 1.077280I
b = −0.363802 − 0.595725I
−7.09081 + 0.79638I 0
u = 0.269938 + 0.288721I
a = −2.52869 + 1.49962I
b = 0.887931 + 0.459497I
2.64104 + 0.25879I 3.85203 + 2.96045I
u = 0.269938 − 0.288721I
a = −2.52869 − 1.49962I
b = 0.887931 − 0.459497I
2.64104 − 0.25879I 3.85203 − 2.96045I
u = 1.61612 + 0.17777I
a = 1.092350 − 0.316288I
b = −0.985940 − 0.438836I
−8.99388 − 4.78654I 0
u = 1.61612 − 0.17777I
a = 1.092350 + 0.316288I
b = −0.985940 + 0.438836I
−8.99388 + 4.78654I 0
u = −1.63927 + 0.09770I
a = 2.05679 − 0.27434I
b = −1.51522 + 0.94459I
−11.15790 + 5.83644I 0
u = −1.63927 − 0.09770I
a = 2.05679 + 0.27434I
b = −1.51522 − 0.94459I
−11.15790 − 5.83644I 0
u = −1.65031 + 0.16673I
a = −1.89221 + 0.02470I
b = 1.30041 − 0.93941I
−7.8166 + 12.4315I 0
u = −1.65031 − 0.16673I
a = −1.89221 − 0.02470I
b = 1.30041 + 0.93941I
−7.8166 − 12.4315I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.67671 + 0.06666I
a = −1.267240 + 0.207888I
b = 0.892941 + 0.200725I
−10.51010 − 0.53898I 0
u = 1.67671 − 0.06666I
a = −1.267240 − 0.207888I
b = 0.892941 − 0.200725I
−10.51010 + 0.53898I 0
8
II. I
u
2
= hb, a + 1, u + 1i
(i) Arc colorings
a
1
=
0
−1
a
5
=
1
0
a
4
=
1
−1
a
8
=
−1
0
a
7
=
−1
0
a
2
=
−1
−1
a
6
=
0
−1
a
10
=
−1
0
a
3
=
0
−1
a
9
=
−1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
u + 1
c
7
u
c
8
, c
9
, c
10
u − 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
8
, c
9
, c
10
y −1
c
7
y
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
0 0
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)(u
32
+ 2u
31
+ ··· + 12u + 8)
c
2
(u + 1)(u
32
+ 11u
30
+ ··· + 13u − 1)
c
3
, c
4
(u + 1)(u
32
+ 2u
31
+ ··· − u + 1)
c
5
(u + 1)(u
32
− 2u
31
+ ··· + u − 1)
c
6
(u + 1)(u
32
+ 2u
31
+ ··· − 13u − 1)
c
7
u(u
32
− 5u
31
+ ··· − 6u + 2)
c
8
(u − 1)(u
32
+ 2u
31
+ ··· − 13u − 1)
c
9
, c
10
(u − 1)(u
32
+ 2u
31
+ ··· − u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
32
+ 30y
31
+ ··· + 240y + 64)
c
2
(y −1)(y
32
+ 22y
31
+ ··· − 121y + 1)
c
3
, c
4
, c
9
c
10
(y −1)(y
32
− 38y
31
+ ··· − 5y + 1)
c
5
(y −1)(y
32
− 6y
31
+ ··· − 5y + 1)
c
6
, c
8
(y −1)(y
32
− 18y
31
+ ··· − 81y + 1)
c
7
y(y
32
− 9y
31
+ ··· − 32y + 4)
14
