10
49
(K10a
13
)
A knot diagram
1
Linearized knot diagam
4 9 5 2 8 10 1 3 6 7
Solving Sequence
6,9
10 7
1,3
2 8 5 4
c
9
c
6
c
10
c
2
c
8
c
5
c
4
c
1
, c
3
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
30
+ 17u
28
+ ··· + b − 1, u
29
+ u
28
+ ··· + a + 1, u
31
+ 2u
30
+ ··· + 2u + 1i
I
u
2
= hb, a − u + 1, u
2
− 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
= h−u
30
+17u
28
+· · ·+b−1, u
29
+u
28
+· · ·+a+1, u
31
+2u
30
+· · ·+2u+1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
u
2
a
7
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
−u
29
− u
28
+ ··· − u − 1
u
30
− 17u
28
+ ··· + 3u + 1
a
2
=
u
30
− u
29
+ ··· − 3u
2
+ 2u
u
30
− 17u
28
+ ··· + 3u + 1
a
8
=
u
3
− 2u
u
5
− 3u
3
+ u
a
5
=
u
7
− 4u
5
+ 4u
3
u
9
− 5u
7
+ 7u
5
− 2u
3
+ u
a
4
=
−u
29
− u
28
+ ··· − u
2
+ u
−u
14
+ 8u
12
+ ··· + u
2
+ 2u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
30
+ 3u
29
−63u
28
−42u
27
+ 425u
26
+ 231u
25
−1601u
24
−582u
23
+ 3678u
22
+ 378u
21
−
5240u
20
+1434u
19
+4267u
18
−3741u
17
−861u
16
+3662u
15
−2258u
14
−1402u
13
+2636u
12
−
406u
11
−1210u
10
+ 852u
9
+ 292u
8
−384u
7
+ 86u
6
+ 142u
5
−32u
4
−6u
3
+ 13u
2
+ 9u −11
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
31
− 3u
30
+ ··· + 3u + 1
c
2
, c
8
u
31
+ u
30
+ ··· + 12u + 4
c
3
u
31
+ 15u
30
+ ··· + 29u + 1
c
5
u
31
− 8u
30
+ ··· + 14u + 7
c
6
, c
7
, c
9
c
10
u
31
+ 2u
30
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
31
− 15y
30
+ ··· + 29y −1
c
2
, c
8
y
31
+ 15y
30
+ ··· − 8y −16
c
3
y
31
+ 5y
30
+ ··· + 505y −1
c
5
y
31
+ 20y
29
+ ··· − 602y −49
c
6
, c
7
, c
9
c
10
y
31
− 36y
30
+ ··· + 10y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.942627 + 0.191065I
a = −0.160124 + 0.103064I
b = 0.324783 + 0.959750I
−1.73768 − 1.98261I −12.51789 + 2.95931I
u = −0.942627 − 0.191065I
a = −0.160124 − 0.103064I
b = 0.324783 − 0.959750I
−1.73768 + 1.98261I −12.51789 − 2.95931I
u = 0.696545 + 0.545292I
a = 1.23592 − 1.62736I
b = 0.613275 + 1.178920I
0.71112 − 8.80296I −11.07196 + 8.43090I
u = 0.696545 − 0.545292I
a = 1.23592 + 1.62736I
b = 0.613275 − 1.178920I
0.71112 + 8.80296I −11.07196 − 8.43090I
u = 0.605327 + 0.533968I
a = −1.15171 + 1.76364I
b = −0.398966 − 1.160740I
2.77360 − 3.43811I −7.57029 + 4.39561I
u = 0.605327 − 0.533968I
a = −1.15171 − 1.76364I
b = −0.398966 + 1.160740I
2.77360 + 3.43811I −7.57029 − 4.39561I
u = −0.605796 + 0.419305I
a = −0.106041 + 0.538372I
b = 0.914628 + 0.393426I
−1.75392 + 3.16934I −13.1405 − 6.2492I
u = −0.605796 − 0.419305I
a = −0.106041 − 0.538372I
b = 0.914628 − 0.393426I
−1.75392 − 3.16934I −13.1405 + 6.2492I
u = 0.216063 + 0.636597I
a = 0.53700 − 1.67610I
b = −0.488198 + 1.161550I
2.12474 + 4.80226I −7.72031 − 3.44347I
u = 0.216063 − 0.636597I
a = 0.53700 + 1.67610I
b = −0.488198 − 1.161550I
2.12474 − 4.80226I −7.72031 + 3.44347I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.331449 + 0.582530I
a = −0.68223 + 1.81325I
b = 0.208622 − 1.161580I
3.57659 − 0.38668I −5.31318 + 2.65084I
u = 0.331449 − 0.582530I
a = −0.68223 − 1.81325I
b = 0.208622 + 1.161580I
3.57659 + 0.38668I −5.31318 − 2.65084I
u = 0.574643 + 0.305412I
a = 1.51598 − 2.33460I
b = 0.313373 + 0.704732I
−2.56499 − 0.98527I −12.14842 + 6.83319I
u = 0.574643 − 0.305412I
a = 1.51598 + 2.33460I
b = 0.313373 − 0.704732I
−2.56499 + 0.98527I −12.14842 − 6.83319I
u = −1.400590 + 0.076803I
a = 0.284363 + 0.723650I
b = 0.076838 − 1.243230I
−1.80597 + 2.68803I −9.99041 − 3.16248I
u = −1.400590 − 0.076803I
a = 0.284363 − 0.723650I
b = 0.076838 + 1.243230I
−1.80597 − 2.68803I −9.99041 + 3.16248I
u = 1.54559 + 0.05817I
a = 0.500839 − 0.189268I
b = 0.922872 − 0.250964I
−7.37189 − 0.63906I −13.31985 + 0.I
u = 1.54559 − 0.05817I
a = 0.500839 + 0.189268I
b = 0.922872 + 0.250964I
−7.37189 + 0.63906I −13.31985 + 0.I
u = −0.283148 + 0.347355I
a = −0.301800 − 0.948705I
b = −0.732891 + 0.139904I
−0.846644 − 0.285966I −10.27924 − 1.27611I
u = −0.283148 − 0.347355I
a = −0.301800 + 0.948705I
b = −0.732891 − 0.139904I
−0.846644 + 0.285966I −10.27924 + 1.27611I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.440544
a = −0.456355
b = −0.447925
−0.703249 −13.8910
u = −1.56849 + 0.15264I
a = 1.094690 + 0.849273I
b = 0.557583 − 1.179560I
−4.51872 + 5.93011I −10.96804 − 3.41229I
u = −1.56849 − 0.15264I
a = 1.094690 − 0.849273I
b = 0.557583 + 1.179560I
−4.51872 − 5.93011I −10.96804 + 3.41229I
u = −1.57353 + 0.09063I
a = −1.14209 − 1.28668I
b = −0.446370 + 0.932965I
−9.92171 + 2.45212I −15.0553 − 2.8825I
u = −1.57353 − 0.09063I
a = −1.14209 + 1.28668I
b = −0.446370 − 0.932965I
−9.92171 − 2.45212I −15.0553 + 2.8825I
u = 1.57590 + 0.11764I
a = −0.484278 + 0.374146I
b = −1.031430 + 0.523808I
−9.15652 − 5.11817I −15.5151 + 3.8713I
u = 1.57590 − 0.11764I
a = −0.484278 − 0.374146I
b = −1.031430 − 0.523808I
−9.15652 + 5.11817I −15.5151 − 3.8713I
u = −1.60251 + 0.16367I
a = −1.24549 − 0.75908I
b = −0.711416 + 1.179640I
−7.05058 + 11.45320I −13.9714 − 7.0213I
u = −1.60251 − 0.16367I
a = −1.24549 + 0.75908I
b = −0.711416 − 1.179640I
−7.05058 − 11.45320I −13.9714 + 7.0213I
u = 1.65145 + 0.04258I
a = −0.166851 + 0.356098I
b = −0.398737 + 0.726247I
−10.63140 + 1.14909I −14.4727 − 5.7136I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.65145 − 0.04258I
a = −0.166851 − 0.356098I
b = −0.398737 − 0.726247I
−10.63140 − 1.14909I −14.4727 + 5.7136I
8
II. I
u
2
= hb, a − u + 1, u
2
− u − 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
u + 1
a
7
=
−u
−u − 1
a
1
=
−u
−u
a
3
=
u − 1
0
a
2
=
u − 1
0
a
8
=
1
0
a
5
=
u
u
a
4
=
2u − 1
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −15
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u − 1)
2
c
2
, c
8
u
2
c
4
(u + 1)
2
c
5
, c
6
, c
7
u
2
+ u − 1
c
9
, c
10
u
2
− u − 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
(y −1)
2
c
2
, c
8
y
2
c
5
, c
6
, c
7
c
9
, c
10
y
2
− 3y + 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −1.61803
b = 0
−2.63189 −15.0000
u = 1.61803
a = 0.618034
b = 0
−10.5276 −15.0000
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
2
)(u
31
− 3u
30
+ ··· + 3u + 1)
c
2
, c
8
u
2
(u
31
+ u
30
+ ··· + 12u + 4)
c
3
((u − 1)
2
)(u
31
+ 15u
30
+ ··· + 29u + 1)
c
4
((u + 1)
2
)(u
31
− 3u
30
+ ··· + 3u + 1)
c
5
(u
2
+ u − 1)(u
31
− 8u
30
+ ··· + 14u + 7)
c
6
, c
7
(u
2
+ u − 1)(u
31
+ 2u
30
+ ··· + 2u + 1)
c
9
, c
10
(u
2
− u − 1)(u
31
+ 2u
30
+ ··· + 2u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
2
)(y
31
− 15y
30
+ ··· + 29y −1)
c
2
, c
8
y
2
(y
31
+ 15y
30
+ ··· − 8y −16)
c
3
((y −1)
2
)(y
31
+ 5y
30
+ ··· + 505y −1)
c
5
(y
2
− 3y + 1)(y
31
+ 20y
29
+ ··· − 602y −49)
c
6
, c
7
, c
9
c
10
(y
2
− 3y + 1)(y
31
− 36y
30
+ ··· + 10y −1)
14
