10
52
(K10a
80
)
A knot diagram
1
Linearized knot diagam
7 10 6 8 9 1 5 4 2 3
Solving Sequence
4,9
8 5 6 3
1,7
2 10
c
8
c
4
c
5
c
3
c
7
c
1
c
10
c
2
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
29
+ u
28
+ ··· + b − 1, u
31
+ 2u
30
+ ··· + a − 3, u
32
+ 2u
31
+ ··· − 5u − 1i
I
u
2
= hb − 1, −u
2
+ a + u − 2, u
3
− u
2
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 35 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
= h2u
29
+u
28
+· · ·+b −1, u
31
+2u
30
+· · ·+a −3, u
32
+2u
31
+· · ·−5u −1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
8
=
1
u
2
a
5
=
u
u
3
+ u
a
6
=
u
3
+ 2u
u
3
+ u
a
3
=
−u
7
− 4u
5
− 4u
3
−u
7
− 3u
5
− 2u
3
+ u
a
1
=
−u
31
− 2u
30
+ ··· + u + 3
−2u
29
− u
28
+ ··· + 2u + 1
a
7
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
−u
31
− 2u
30
+ ··· + 6u + 4
−u
29
− u
28
+ ··· + u + 1
a
10
=
−u
31
− 2u
30
+ ··· + 4u + 4
−u
29
− u
28
+ ··· + 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= u
31
+ 2u
30
+ 16u
29
+ 30u
28
+ 116u
27
+ 201u
26
+ 500u
25
+ 783u
24
+ 1406u
23
+ 1926u
22
+
2640u
21
+ 3005u
20
+ 3188u
19
+ 2713u
18
+ 2078u
17
+ 811u
16
+ 52u
15
−886u
14
−904u
13
−
890u
12
− 388u
11
− 158u
10
+ 98u
9
− 10u
7
− 104u
6
− 38u
5
− 21u
4
+ 42u
3
+ 21u
2
+ 6u − 9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
32
− u
31
+ ··· + 4u + 8
c
2
, c
9
, c
10
u
32
− 4u
31
+ ··· + 2u − 1
c
3
u
32
+ 6u
31
+ ··· − 29u + 19
c
4
, c
7
, c
8
u
32
+ 2u
31
+ ··· − 5u − 1
c
5
u
32
− 2u
31
+ ··· − 91u − 17
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
32
− 21y
31
+ ··· − 400y + 64
c
2
, c
9
, c
10
y
32
− 32y
31
+ ··· + 10y + 1
c
3
y
32
+ 18y
31
+ ··· − 14597y + 361
c
4
, c
7
, c
8
y
32
+ 30y
31
+ ··· − 17y + 1
c
5
y
32
+ 6y
31
+ ··· − 2025y + 289
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.533924 + 0.635384I
a = −1.44678 + 1.16510I
b = −0.03219 + 1.54133I
−8.33717 + 3.47045I −6.19300 − 0.53804I
u = −0.533924 − 0.635384I
a = −1.44678 − 1.16510I
b = −0.03219 − 1.54133I
−8.33717 − 3.47045I −6.19300 + 0.53804I
u = −0.737398 + 0.363177I
a = 1.39466 − 1.92116I
b = 0.33070 − 1.92317I
−7.36935 − 7.82848I −4.18330 + 6.10894I
u = −0.737398 − 0.363177I
a = 1.39466 + 1.92116I
b = 0.33070 + 1.92317I
−7.36935 + 7.82848I −4.18330 − 6.10894I
u = 0.121416 + 1.191480I
a = 0.222642 − 0.520130I
b = −0.646759 − 0.202123I
−1.84659 + 2.03195I −0.06352 − 4.09496I
u = 0.121416 − 1.191480I
a = 0.222642 + 0.520130I
b = −0.646759 + 0.202123I
−1.84659 − 2.03195I −0.06352 + 4.09496I
u = 0.772369
a = −0.383393
b = 0.296121
−2.56303 −3.36180
u = 0.321817 + 1.204360I
a = −0.378231 + 0.177725I
b = 0.335766 + 0.398330I
−6.26853 + 3.96490I −7.15642 − 4.13069I
u = 0.321817 − 1.204360I
a = −0.378231 − 0.177725I
b = 0.335766 − 0.398330I
−6.26853 − 3.96490I −7.15642 + 4.13069I
u = −0.046033 + 1.276630I
a = 0.169895 + 1.097880I
b = 1.40941 − 0.16635I
−4.89788 − 1.11555I −6.11098 − 0.26189I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.046033 − 1.276630I
a = 0.169895 − 1.097880I
b = 1.40941 + 0.16635I
−4.89788 + 1.11555I −6.11098 + 0.26189I
u = −0.637579 + 0.336310I
a = −1.77319 + 1.89857I
b = −0.49204 + 1.80683I
−1.10997 − 4.05552I −1.42840 + 6.80075I
u = −0.637579 − 0.336310I
a = −1.77319 − 1.89857I
b = −0.49204 − 1.80683I
−1.10997 + 4.05552I −1.42840 − 6.80075I
u = 0.573185 + 0.380549I
a = −0.567444 − 0.158963I
b = 0.264757 + 0.307056I
−3.48280 + 1.78898I −3.34736 − 3.66370I
u = 0.573185 − 0.380549I
a = −0.567444 + 0.158963I
b = 0.264757 − 0.307056I
−3.48280 − 1.78898I −3.34736 + 3.66370I
u = 0.214793 + 1.351600I
a = 0.225799 + 0.123979I
b = 0.119070 − 0.331821I
−3.45767 + 3.36417I 0.37870 − 3.50479I
u = 0.214793 − 1.351600I
a = 0.225799 − 0.123979I
b = 0.119070 + 0.331821I
−3.45767 − 3.36417I 0.37870 + 3.50479I
u = −0.457656 + 0.423798I
a = 1.94595 − 1.20331I
b = 0.38062 − 1.37539I
−1.72217 + 0.51232I −4.14141 + 0.14369I
u = −0.457656 − 0.423798I
a = 1.94595 + 1.20331I
b = 0.38062 + 1.37539I
−1.72217 − 0.51232I −4.14141 − 0.14369I
u = 0.569557 + 0.125662I
a = 0.316122 + 0.218549I
b = −0.152586 − 0.164201I
1.244440 + 0.519638I 6.41959 − 1.56914I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.569557 − 0.125662I
a = 0.316122 − 0.218549I
b = −0.152586 + 0.164201I
1.244440 − 0.519638I 6.41959 + 1.56914I
u = −0.19027 + 1.43367I
a = 1.50108 + 0.51525I
b = 1.02431 − 2.05401I
−7.59173 − 1.96238I −7.59391 + 0.I
u = −0.19027 − 1.43367I
a = 1.50108 − 0.51525I
b = 1.02431 + 2.05401I
−7.59173 + 1.96238I −7.59391 + 0.I
u = −0.24454 + 1.43301I
a = −1.68707 − 0.19420I
b = −0.69085 + 2.37010I
−6.78693 − 7.28997I −5.63030 + 6.08966I
u = −0.24454 − 1.43301I
a = −1.68707 + 0.19420I
b = −0.69085 − 2.37010I
−6.78693 + 7.28997I −5.63030 − 6.08966I
u = 0.21981 + 1.44034I
a = −0.396703 − 0.147942I
b = −0.125890 + 0.603905I
−9.32026 + 4.72345I −7.29654 − 3.13438I
u = 0.21981 − 1.44034I
a = −0.396703 + 0.147942I
b = −0.125890 − 0.603905I
−9.32026 − 4.72345I −7.29654 + 3.13438I
u = −0.28148 + 1.45411I
a = 1.58198 − 0.01901I
b = 0.41766 − 2.30572I
−13.2076 − 11.5375I −7.79347 + 6.25344I
u = −0.28148 − 1.45411I
a = 1.58198 + 0.01901I
b = 0.41766 + 2.30572I
−13.2076 + 11.5375I −7.79347 − 6.25344I
u = −0.14244 + 1.49315I
a = −1.134180 − 0.397538I
b = −0.75514 + 1.63687I
−15.2480 + 1.1861I −9.66994 + 0.I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.14244 − 1.49315I
a = −1.134180 + 0.397538I
b = −0.75514 − 1.63687I
−15.2480 − 1.1861I −9.66994 + 0.I
u = −0.270853
a = 3.43431
b = 0.930194
−1.22025 −10.0180
8
II. I
u
2
= hb − 1, −u
2
+ a + u − 2, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
8
=
1
u
2
a
5
=
u
u
2
− u + 1
a
6
=
u
2
+ 1
u
2
− u + 1
a
3
=
−1
0
a
1
=
u
2
− u + 2
1
a
7
=
u
2
+ 1
u
2
− u + 1
a
2
=
u
2
− u + 2
1
a
10
=
u
2
− u + 3
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
2
− 4u + 4
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
3
c
2
(u + 1)
3
c
3
, c
5
u
3
− u
2
+ 1
c
4
u
3
+ u
2
+ 2u + 1
c
7
, c
8
u
3
− u
2
+ 2u − 1
c
9
, c
10
(u − 1)
3
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
3
c
2
, c
9
, c
10
(y −1)
3
c
3
, c
5
y
3
− y
2
+ 2y −1
c
4
, c
7
, c
8
y
3
+ 3y
2
+ 2y −1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = 0.122561 − 0.744862I
b = 1.00000
−4.66906 + 2.82812I −5.17211 − 2.41717I
u = 0.215080 − 1.307140I
a = 0.122561 + 0.744862I
b = 1.00000
−4.66906 − 2.82812I −5.17211 + 2.41717I
u = 0.569840
a = 1.75488
b = 1.00000
−0.531480 3.34420
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
u
3
(u
32
− u
31
+ ··· + 4u + 8)
c
2
((u + 1)
3
)(u
32
− 4u
31
+ ··· + 2u − 1)
c
3
(u
3
− u
2
+ 1)(u
32
+ 6u
31
+ ··· − 29u + 19)
c
4
(u
3
+ u
2
+ 2u + 1)(u
32
+ 2u
31
+ ··· − 5u − 1)
c
5
(u
3
− u
2
+ 1)(u
32
− 2u
31
+ ··· − 91u − 17)
c
7
, c
8
(u
3
− u
2
+ 2u − 1)(u
32
+ 2u
31
+ ··· − 5u − 1)
c
9
, c
10
((u − 1)
3
)(u
32
− 4u
31
+ ··· + 2u − 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
3
(y
32
− 21y
31
+ ··· − 400y + 64)
c
2
, c
9
, c
10
((y −1)
3
)(y
32
− 32y
31
+ ··· + 10y + 1)
c
3
(y
3
− y
2
+ 2y −1)(y
32
+ 18y
31
+ ··· − 14597y + 361)
c
4
, c
7
, c
8
(y
3
+ 3y
2
+ 2y −1)(y
32
+ 30y
31
+ ··· − 17y + 1)
c
5
(y
3
− y
2
+ 2y −1)(y
32
+ 6y
31
+ ··· − 2025y + 289)
14
