10
72
(K10a
4
)
A knot diagram
1
Linearized knot diagam
8 6 7 9 4 3 10 1 5 2
Solving Sequence
2,6
3 7
4,8
1 5 10 9
c
2
c
6
c
3
c
1
c
5
c
10
c
9
c
4
, c
7
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
37
− 2u
36
+ ··· + 2b + 1, −u
12
+ 5u
10
+ 2u
9
− 9u
8
− 8u
7
+ 4u
6
+ 10u
5
+ 6u
4
− 2u
3
− 5u
2
+ a − 2u − 1,
u
38
− 3u
37
+ ··· + 2u − 1i
I
u
2
= hb
2
− b + 1, a + 1, u − 1i
* 2 irreducible components of dim
C
= 0, with total 40 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
= hu
37
−2u
36
+· · ·+2b+1, −u
12
+5u
10
+· · ·+a−1, u
38
−3u
37
+· · ·+2u−1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
7
=
−u
−u
3
+ u
a
4
=
−u
2
+ 1
−u
4
+ 2u
2
a
8
=
u
12
− 5u
10
− 2u
9
+ 9u
8
+ 8u
7
− 4u
6
− 10u
5
− 6u
4
+ 2u
3
+ 5u
2
+ 2u + 1
−
1
2
u
37
+ u
36
+ ··· +
5
2
u −
1
2
a
1
=
−
1
2
u
37
+ u
36
+ ··· +
5
2
u +
1
2
5
2
u
37
− 4u
36
+ ··· −
5
2
u +
3
2
a
5
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
10
=
2u
37
− 3u
36
+ ··· + 10u
2
+ 2
5
2
u
37
− 4u
36
+ ··· −
5
2
u +
3
2
a
9
=
−3u
37
+ 5u
36
+ ··· + 5u − 2
3
2
u
37
− 2u
36
+ ··· −
1
2
u +
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
37
−u
36
+35u
35
+21u
34
−270u
33
−205u
32
+1194u
31
+1182u
30
−3222u
29
−4364u
28
+
4873u
27
+ 10507u
26
−1460u
25
−15693u
24
−9964u
23
+ 11016u
22
+ 21182u
21
+ 5747u
20
−
16959u
19
−19349u
18
−2018u
17
+ 13493u
16
+ 13726u
15
+ 2674u
14
−6790u
13
−7586u
12
−
3020u
11
+1300u
10
+2786u
9
+1852u
8
+442u
7
−314u
6
−448u
5
−263u
4
−91u
3
−25u
2
+4u−5
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
38
+ 2u
37
+ ··· + 5u + 1
c
2
, c
3
, c
6
u
38
− 3u
37
+ ··· + 2u − 1
c
4
, c
9
u
38
− u
37
+ ··· − 4u − 4
c
5
u
38
+ 15u
37
+ ··· + 72u + 16
c
7
u
38
− 2u
37
+ ··· − 37u + 17
c
10
u
38
+ 18u
37
+ ··· − 5u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
38
+ 18y
37
+ ··· − 5y + 1
c
2
, c
3
, c
6
y
38
− 33y
37
+ ··· − 8y + 1
c
4
, c
9
y
38
− 15y
37
+ ··· − 72y + 16
c
5
y
38
+ 13y
37
+ ··· − 2848y + 256
c
7
y
38
− 6y
37
+ ··· − 6333y + 289
c
10
y
38
+ 6y
37
+ ··· − 61y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.000620 + 0.466336I
a = 0.694761 − 0.337574I
b = −0.527298 + 1.065360I
−2.40471 − 3.95746I −10.27520 + 4.57056I
u = −1.000620 − 0.466336I
a = 0.694761 + 0.337574I
b = −0.527298 − 1.065360I
−2.40471 + 3.95746I −10.27520 − 4.57056I
u = −1.062270 + 0.332916I
a = 1.252160 − 0.030949I
b = −0.570085 − 0.447308I
−0.568983 + 0.479860I −6.06539 + 0.48126I
u = −1.062270 − 0.332916I
a = 1.252160 + 0.030949I
b = −0.570085 + 0.447308I
−0.568983 − 0.479860I −6.06539 − 0.48126I
u = −0.719303 + 0.499357I
a = 1.44677 − 0.15075I
b = −0.362704 − 1.048010I
−3.54227 + 2.75914I −13.19764 − 4.35912I
u = −0.719303 − 0.499357I
a = 1.44677 + 0.15075I
b = −0.362704 + 1.048010I
−3.54227 − 2.75914I −13.19764 + 4.35912I
u = −0.214521 + 0.842165I
a = 2.14112 + 0.49356I
b = −0.573770 − 1.100590I
0.00579 + 8.62980I −6.60829 − 7.80256I
u = −0.214521 − 0.842165I
a = 2.14112 − 0.49356I
b = −0.573770 + 1.100590I
0.00579 − 8.62980I −6.60829 + 7.80256I
u = −0.174468 + 0.788088I
a = 1.39685 − 0.95450I
b = −0.731729 + 0.388434I
2.09675 + 3.65224I −3.04639 − 3.74887I
u = −0.174468 − 0.788088I
a = 1.39685 + 0.95450I
b = −0.731729 − 0.388434I
2.09675 − 3.65224I −3.04639 + 3.74887I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.317784 + 0.691757I
a = 0.058360 − 0.565761I
b = −0.214760 + 1.058960I
−2.35641 + 1.43399I −10.35352 − 2.88902I
u = −0.317784 − 0.691757I
a = 0.058360 + 0.565761I
b = −0.214760 − 1.058960I
−2.35641 − 1.43399I −10.35352 + 2.88902I
u = 1.251800 + 0.201783I
a = −0.999363 − 0.281232I
b = 0.652278 + 0.954226I
−2.15443 + 0.39089I −9.84825 + 1.14697I
u = 1.251800 − 0.201783I
a = −0.999363 + 0.281232I
b = 0.652278 − 0.954226I
−2.15443 − 0.39089I −9.84825 − 1.14697I
u = −1.260340 + 0.253559I
a = −0.003037 + 0.946509I
b = 0.662945 − 0.361405I
−1.02515 + 1.90334I −5.81979 − 1.07076I
u = −1.260340 − 0.253559I
a = −0.003037 − 0.946509I
b = 0.662945 + 0.361405I
−1.02515 − 1.90334I −5.81979 + 1.07076I
u = 1.284790 + 0.261207I
a = −1.278000 + 0.104536I
b = 0.731652 − 0.644131I
−1.23963 − 4.86305I −7.46881 + 6.13263I
u = 1.284790 − 0.261207I
a = −1.278000 − 0.104536I
b = 0.731652 + 0.644131I
−1.23963 + 4.86305I −7.46881 − 6.13263I
u = −0.016707 + 0.678781I
a = −1.38023 − 1.35662I
b = 0.666801 + 0.530991I
2.80379 + 1.46931I −1.12935 − 3.08473I
u = −0.016707 − 0.678781I
a = −1.38023 + 1.35662I
b = 0.666801 − 0.530991I
2.80379 − 1.46931I −1.12935 + 3.08473I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.315510 + 0.121395I
a = 0.61655 − 1.44009I
b = 0.291142 − 1.061280I
−4.89567 − 0.49664I −12.27278 + 1.11503I
u = −1.315510 − 0.121395I
a = 0.61655 + 1.44009I
b = 0.291142 + 1.061280I
−4.89567 + 0.49664I −12.27278 − 1.11503I
u = −1.329280 + 0.259672I
a = −1.99856 + 1.39234I
b = 0.547737 + 1.093970I
−3.13917 + 6.61979I −9.45062 − 5.39938I
u = −1.329280 − 0.259672I
a = −1.99856 − 1.39234I
b = 0.547737 − 1.093970I
−3.13917 − 6.61979I −9.45062 + 5.39938I
u = 0.091958 + 0.636482I
a = −2.64958 + 0.37806I
b = 0.571517 − 1.023410I
1.34864 − 3.34557I −3.46602 + 2.94107I
u = 0.091958 − 0.636482I
a = −2.64958 − 0.37806I
b = 0.571517 + 1.023410I
1.34864 + 3.34557I −3.46602 − 2.94107I
u = 1.372870 + 0.330158I
a = 0.451776 + 0.876637I
b = −0.811572 − 0.358412I
−2.79986 − 7.69321I 0
u = 1.372870 − 0.330158I
a = 0.451776 − 0.876637I
b = −0.811572 + 0.358412I
−2.79986 + 7.69321I 0
u = −0.582954
a = 0.891810
b = −0.340706
−0.970134 −9.92360
u = 1.42045
a = 0.242047
b = −0.758415
−7.33419 −11.4900
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.40646 + 0.27352I
a = −0.517124 − 0.625628I
b = −0.183991 − 1.166970I
−7.80695 − 4.93169I 0
u = 1.40646 − 0.27352I
a = −0.517124 + 0.625628I
b = −0.183991 + 1.166970I
−7.80695 + 4.93169I 0
u = 1.39814 + 0.35135I
a = 1.96262 + 0.82883I
b = −0.591496 + 1.133850I
−5.10692 − 12.92960I 0
u = 1.39814 − 0.35135I
a = 1.96262 − 0.82883I
b = −0.591496 − 1.133850I
−5.10692 + 12.92960I 0
u = 1.47061 + 0.05198I
a = 0.671770 + 1.172880I
b = −0.422515 + 1.169490I
−10.78740 − 4.17106I 0
u = 1.47061 − 0.05198I
a = 0.671770 − 1.172880I
b = −0.422515 − 1.169490I
−10.78740 + 4.17106I 0
u = 0.215436 + 0.157466I
a = 1.56623 + 0.67273I
b = 0.415410 + 0.878457I
−0.33342 + 1.74546I −2.32569 − 3.49934I
u = 0.215436 − 0.157466I
a = 1.56623 − 0.67273I
b = 0.415410 − 0.878457I
−0.33342 − 1.74546I −2.32569 + 3.49934I
8
II. I
u
2
= hb
2
− b + 1, a + 1, u − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
3
=
1
1
a
7
=
−1
0
a
4
=
0
1
a
8
=
−1
b
a
1
=
−b + 1
b − 1
a
5
=
0
1
a
10
=
0
b − 1
a
9
=
0
b − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4b − 7
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
10
u
2
− u + 1
c
2
, c
3
(u − 1)
2
c
4
, c
5
, c
9
u
2
c
6
(u + 1)
2
c
8
u
2
+ u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
8
c
10
y
2
+ y + 1
c
2
, c
3
, c
6
(y −1)
2
c
4
, c
5
, c
9
y
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = 0.500000 + 0.866025I
−1.64493 + 2.02988I −9.00000 − 3.46410I
u = 1.00000
a = −1.00000
b = 0.500000 − 0.866025I
−1.64493 − 2.02988I −9.00000 + 3.46410I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)(u
38
+ 2u
37
+ ··· + 5u + 1)
c
2
, c
3
((u − 1)
2
)(u
38
− 3u
37
+ ··· + 2u − 1)
c
4
, c
9
u
2
(u
38
− u
37
+ ··· − 4u − 4)
c
5
u
2
(u
38
+ 15u
37
+ ··· + 72u + 16)
c
6
((u + 1)
2
)(u
38
− 3u
37
+ ··· + 2u − 1)
c
7
(u
2
− u + 1)(u
38
− 2u
37
+ ··· − 37u + 17)
c
8
(u
2
+ u + 1)(u
38
+ 2u
37
+ ··· + 5u + 1)
c
10
(u
2
− u + 1)(u
38
+ 18u
37
+ ··· − 5u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y
2
+ y + 1)(y
38
+ 18y
37
+ ··· − 5y + 1)
c
2
, c
3
, c
6
((y −1)
2
)(y
38
− 33y
37
+ ··· − 8y + 1)
c
4
, c
9
y
2
(y
38
− 15y
37
+ ··· − 72y + 16)
c
5
y
2
(y
38
+ 13y
37
+ ··· − 2848y + 256)
c
7
(y
2
+ y + 1)(y
38
− 6y
37
+ ··· − 6333y + 289)
c
10
(y
2
+ y + 1)(y
38
+ 6y
37
+ ··· − 61y + 1)
14
