10
80
(K10a
8
)
A knot diagram
1
Linearized knot diagam
9 6 10 7 3 8 5 1 2 4
Solving Sequence
4,7
5
1,8
9 6 10 3 2
c
4
c
7
c
8
c
6
c
10
c
3
c
2
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h1794841722415u
39
+ 5490595544415u
38
+ ··· + 1305995790962b + 2649665691745,
− 1190941729941u
39
− 4615935344485u
38
+ ··· + 1305995790962a − 3386435539405,
u
40
+ 4u
39
+ ··· − 2u + 1i
I
u
2
= hb, u
2
+ a + 2u + 1, u
3
+ u
2
− 1i
I
u
3
= hb − a − 1, a
2
+ a − 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 45 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
12
u
39
+5.49×10
12
u
38
+· · ·+1.31×10
12
b+2.65×10
12
, −1.19×
10
12
u
39
−4.62×10
12
u
38
+· · ·+1.31×10
12
a−3.39×10
12
, u
40
+4u
39
+· · ·−2u+1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
5
=
1
u
2
a
1
=
0.911903u
39
+ 3.53442u
38
+ ··· + 4.75622u + 2.59299
−1.37431u
39
− 4.20414u
38
+ ··· + 3.79441u − 2.02885
a
8
=
−u
−u
3
+ u
a
9
=
−2.36296u
39
− 8.30641u
38
+ ··· + 1.11511u − 2.81278
0.625691u
39
+ 1.79586u
38
+ ··· − 1.20559u + 0.971153
a
6
=
u
3
u
5
− u
3
+ u
a
10
=
−0.462406u
39
− 0.669727u
38
+ ··· + 8.55064u + 0.564144
−1.37431u
39
− 4.20414u
38
+ ··· + 3.79441u − 2.02885
a
3
=
−0.481242u
39
− 1.19932u
38
+ ··· + 3.81151u + 1.61679
−1.34743u
39
− 3.78400u
38
+ ··· + 3.42400u − 1.70725
a
2
=
−2.58732u
39
− 8.51193u
38
+ ··· + 10.6671u − 0.875683
−0.625691u
39
− 1.79586u
38
+ ··· + 1.20559u − 0.971153
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
1683630050026
652997895481
u
39
−
6742803896956
652997895481
u
38
+ ··· −
5009068136946
652997895481
u −
5965220685850
652997895481
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
, c
9
u
40
− 5u
39
+ ··· + 6u + 1
c
2
, c
5
u
40
− 2u
39
+ ··· + 4u − 4
c
3
, c
10
u
40
+ 2u
39
+ ··· − 28u − 8
c
4
, c
7
u
40
− 4u
39
+ ··· + 2u + 1
c
6
u
40
+ 20u
39
+ ··· + 38u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
, c
9
y
40
− 39y
39
+ ··· + 24y + 1
c
2
, c
5
y
40
+ 18y
39
+ ··· − 104y + 16
c
3
, c
10
y
40
− 24y
39
+ ··· − 1360y + 64
c
4
, c
7
y
40
− 20y
39
+ ··· − 38y + 1
c
6
y
40
+ 4y
39
+ ··· − 918y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.272416 + 0.968858I
a = 0.945891 + 0.854682I
b = −1.224160 − 0.640097I
−3.73330 − 7.68923I −11.60024 + 4.76581I
u = −0.272416 − 0.968858I
a = 0.945891 − 0.854682I
b = −1.224160 + 0.640097I
−3.73330 + 7.68923I −11.60024 − 4.76581I
u = −0.645648 + 0.698758I
a = 0.141845 + 1.079620I
b = 0.488954 − 0.746861I
3.38024 + 1.21441I −4.33120 − 2.38202I
u = −0.645648 − 0.698758I
a = 0.141845 − 1.079620I
b = 0.488954 + 0.746861I
3.38024 − 1.21441I −4.33120 + 2.38202I
u = −1.038880 + 0.250251I
a = 0.485795 + 0.350283I
b = 1.66075 + 0.19671I
−10.88700 + 0.63545I −16.1019 − 7.3224I
u = −1.038880 − 0.250251I
a = 0.485795 − 0.350283I
b = 1.66075 − 0.19671I
−10.88700 − 0.63545I −16.1019 + 7.3224I
u = 0.917670
a = 4.22167
b = 0.349359
−2.98695 −59.3920
u = 1.033250 + 0.435364I
a = −0.60097 + 1.79230I
b = −0.986819 − 0.340805I
−2.52882 − 3.14028I −13.2871 + 4.9220I
u = 1.033250 − 0.435364I
a = −0.60097 − 1.79230I
b = −0.986819 + 0.340805I
−2.52882 + 3.14028I −13.2871 − 4.9220I
u = 0.424088 + 0.764374I
a = −1.58059 + 0.54433I
b = 1.213240 − 0.287237I
−6.42531 + 1.37910I −14.4871 − 0.1126I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.424088 − 0.764374I
a = −1.58059 − 0.54433I
b = 1.213240 + 0.287237I
−6.42531 − 1.37910I −14.4871 + 0.1126I
u = −0.334699 + 0.793502I
a = −0.423088 − 0.639117I
b = 1.031810 + 0.544946I
1.73108 − 3.69196I −7.37427 + 4.06105I
u = −0.334699 − 0.793502I
a = −0.423088 + 0.639117I
b = 1.031810 − 0.544946I
1.73108 + 3.69196I −7.37427 − 4.06105I
u = 1.096340 + 0.338707I
a = −1.05201 + 1.21469I
b = 0.110133 − 0.969437I
−4.81110 − 1.15004I −14.8249 + 0.1630I
u = 1.096340 − 0.338707I
a = −1.05201 − 1.21469I
b = 0.110133 + 0.969437I
−4.81110 + 1.15004I −14.8249 − 0.1630I
u = −0.955160 + 0.637303I
a = −0.565833 − 0.448992I
b = 0.220904 + 0.771822I
2.47440 + 3.90124I −5.43445 − 4.68146I
u = −0.955160 − 0.637303I
a = −0.565833 + 0.448992I
b = 0.220904 − 0.771822I
2.47440 − 3.90124I −5.43445 + 4.68146I
u = −1.048110 + 0.492760I
a = −0.904373 − 0.926403I
b = −1.239580 + 0.203806I
−2.11016 + 3.32020I −13.06049 − 3.76837I
u = −1.048110 − 0.492760I
a = −0.904373 + 0.926403I
b = −1.239580 − 0.203806I
−2.11016 − 3.32020I −13.06049 + 3.76837I
u = 1.160490 + 0.215401I
a = 1.009060 − 0.552611I
b = 0.895187 − 0.176420I
−3.04518 + 0.83928I −13.6876 − 5.4055I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.160490 − 0.215401I
a = 1.009060 + 0.552611I
b = 0.895187 + 0.176420I
−3.04518 − 0.83928I −13.6876 + 5.4055I
u = −1.109770 + 0.525691I
a = 0.749622 + 0.680872I
b = −0.374879 − 1.281230I
−3.50534 + 6.28261I −13.0953 − 5.4809I
u = −1.109770 − 0.525691I
a = 0.749622 − 0.680872I
b = −0.374879 + 1.281230I
−3.50534 − 6.28261I −13.0953 + 5.4809I
u = −0.873586 + 0.885492I
a = 0.722061 − 0.543762I
b = −0.840743 + 0.122050I
0.49614 + 3.22180I −15.2960 − 4.0561I
u = −0.873586 − 0.885492I
a = 0.722061 + 0.543762I
b = −0.840743 − 0.122050I
0.49614 − 3.22180I −15.2960 + 4.0561I
u = 1.109100 + 0.586635I
a = −0.04572 − 1.84175I
b = 1.281130 + 0.518288I
−8.48566 − 6.50843I −15.7623 + 4.5910I
u = 1.109100 − 0.586635I
a = −0.04572 + 1.84175I
b = 1.281130 − 0.518288I
−8.48566 + 6.50843I −15.7623 − 4.5910I
u = −1.135680 + 0.577352I
a = 0.73159 + 1.41035I
b = 1.232290 − 0.518147I
−0.64027 + 8.82354I −11.18744 − 7.65851I
u = −1.135680 − 0.577352I
a = 0.73159 − 1.41035I
b = 1.232290 + 0.518147I
−0.64027 − 8.82354I −11.18744 + 7.65851I
u = 0.684183 + 0.185929I
a = 1.011580 − 0.590171I
b = −0.399719 + 0.274052I
−0.945608 − 0.085520I −9.49008 − 0.83288I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.684183 − 0.185929I
a = 1.011580 + 0.590171I
b = −0.399719 − 0.274052I
−0.945608 + 0.085520I −9.49008 + 0.83288I
u = −0.289056 + 0.640853I
a = −0.58487 − 1.93617I
b = −0.435741 + 0.971160I
−1.17381 − 1.71654I −9.22754 + 1.14237I
u = −0.289056 − 0.640853I
a = −0.58487 + 1.93617I
b = −0.435741 − 0.971160I
−1.17381 + 1.71654I −9.22754 − 1.14237I
u = −0.491493 + 0.483729I
a = −0.533644 + 0.146067I
b = −0.888256 − 0.454789I
−0.414732 + 0.767581I −9.73697 − 1.10255I
u = −0.491493 − 0.483729I
a = −0.533644 − 0.146067I
b = −0.888256 + 0.454789I
−0.414732 − 0.767581I −9.73697 + 1.10255I
u = −1.217970 + 0.609804I
a = −0.43882 − 1.60438I
b = −1.33819 + 0.73038I
−6.6238 + 13.3940I −14.1442 − 7.8976I
u = −1.217970 − 0.609804I
a = −0.43882 + 1.60438I
b = −1.33819 − 0.73038I
−6.6238 − 13.3940I −14.1442 + 7.8976I
u = 1.355550 + 0.252070I
a = −0.147210 + 0.232788I
b = −1.289140 + 0.415642I
−9.24274 + 3.54815I −15.7354 − 3.2017I
u = 1.355550 − 0.252070I
a = −0.147210 − 0.232788I
b = −1.289140 − 0.415642I
−9.24274 − 3.54815I −15.7354 + 3.2017I
u = 0.181281
a = 2.93770
b = −0.583695
−0.821503 −11.8790
8
II. I
u
2
= hb, u
2
+ a + 2u + 1, u
3
+ u
2
− 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
5
=
1
u
2
a
1
=
−u
2
− 2u − 1
0
a
8
=
−u
u
2
+ u − 1
a
9
=
−u
2
− 3u − 1
u
2
+ u − 1
a
6
=
−u
2
+ 1
u
2
a
10
=
−u
2
− 2u − 1
0
a
3
=
1
0
a
2
=
u
−u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
− 8
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u + 1)
3
c
2
, c
6
u
3
− u
2
+ 2u − 1
c
3
, c
10
u
3
c
4
u
3
+ u
2
− 1
c
5
u
3
+ u
2
+ 2u + 1
c
7
u
3
− u
2
+ 1
c
8
, c
9
(u − 1)
3
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
, c
9
(y −1)
3
c
2
, c
5
, c
6
y
3
+ 3y
2
+ 2y −1
c
3
, c
10
y
3
c
4
, c
7
y
3
− y
2
+ 2y −1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = 0.539798 − 0.182582I
b = 0
1.37919 + 2.82812I −7.78492 − 1.30714I
u = −0.877439 − 0.744862I
a = 0.539798 + 0.182582I
b = 0
1.37919 − 2.82812I −7.78492 + 1.30714I
u = 0.754878
a = −3.07960
b = 0
−2.75839 −7.43020
12
III. I
u
3
= hb − a − 1, a
2
+ a − 1, u − 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
1
a
5
=
1
1
a
1
=
a
a + 1
a
8
=
−1
0
a
9
=
−2
−a − 2
a
6
=
1
1
a
10
=
2a + 1
a + 1
a
3
=
−a − 2
−a − 2
a
2
=
−a − 2
−a − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −11
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
2
− u − 1
c
2
, c
5
u
2
c
4
, c
6
(u − 1)
2
c
7
(u + 1)
2
c
8
, c
9
, c
10
u
2
+ u − 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
8
c
9
, c
10
y
2
− 3y + 1
c
2
, c
5
y
2
c
4
, c
6
, c
7
(y −1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.618034
b = 1.61803
−10.5276 −11.0000
u = 1.00000
a = −1.61803
b = −0.618034
−2.63189 −11.0000
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u + 1)
3
)(u
2
− u − 1)(u
40
− 5u
39
+ ··· + 6u + 1)
c
2
u
2
(u
3
− u
2
+ 2u − 1)(u
40
− 2u
39
+ ··· + 4u − 4)
c
3
u
3
(u
2
− u − 1)(u
40
+ 2u
39
+ ··· − 28u − 8)
c
4
((u − 1)
2
)(u
3
+ u
2
− 1)(u
40
− 4u
39
+ ··· + 2u + 1)
c
5
u
2
(u
3
+ u
2
+ 2u + 1)(u
40
− 2u
39
+ ··· + 4u − 4)
c
6
((u − 1)
2
)(u
3
− u
2
+ 2u − 1)(u
40
+ 20u
39
+ ··· + 38u + 1)
c
7
((u + 1)
2
)(u
3
− u
2
+ 1)(u
40
− 4u
39
+ ··· + 2u + 1)
c
8
, c
9
((u − 1)
3
)(u
2
+ u − 1)(u
40
− 5u
39
+ ··· + 6u + 1)
c
10
u
3
(u
2
+ u − 1)(u
40
+ 2u
39
+ ··· − 28u − 8)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
, c
9
((y −1)
3
)(y
2
− 3y + 1)(y
40
− 39y
39
+ ··· + 24y + 1)
c
2
, c
5
y
2
(y
3
+ 3y
2
+ 2y −1)(y
40
+ 18y
39
+ ··· − 104y + 16)
c
3
, c
10
y
3
(y
2
− 3y + 1)(y
40
− 24y
39
+ ··· − 1360y + 64)
c
4
, c
7
((y −1)
2
)(y
3
− y
2
+ 2y −1)(y
40
− 20y
39
+ ··· − 38y + 1)
c
6
((y −1)
2
)(y
3
+ 3y
2
+ 2y −1)(y
40
+ 4y
39
+ ··· − 918y + 1)
18
