10
87
(K10a
39
)
A knot diagram
1
Linearized knot diagam
6 7 1 8 4 10 9 5 2 3
Solving Sequence
4,8
5 6
1,9
3 7 2 10
c
4
c
5
c
8
c
3
c
7
c
2
c
10
c
1
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h2225248116121u
40
− 2393989479567u
39
+ ··· + 10297376929134b − 6012630756121,
− 31071718506119u
40
+ 39367206545589u
39
+ ··· + 5148688464567a − 44602733573152,
u
41
− 2u
40
+ ··· − u − 1i
I
u
2
= hb − 1, a − 1, u − 1i
* 2 irreducible components of dim
C
= 0, with total 42 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
= h2.23×10
12
u
40
−2.39×10
12
u
39
+· · ·+1.03×10
13
b−6.01×10
12
, −3.11×
10
13
u
40
+3.94×10
13
u
39
+· · ·+5.15×10
12
a−4.46×10
13
, u
41
−2u
40
+· · ·−u−1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
5
=
1
u
2
a
6
=
−u
2
+ 1
u
2
a
1
=
6.03488u
40
− 7.64606u
39
+ ··· + 20.1260u + 8.66293
−0.216099u
40
+ 0.232485u
39
+ ··· − 1.20415u + 0.583899
a
9
=
−u
−u
3
+ u
a
3
=
5.64438u
40
− 7.18325u
39
+ ··· + 18.2918u + 8.93217
−0.135606u
40
+ 0.0700585u
39
+ ··· − 1.18339u + 0.664403
a
7
=
u
3
u
5
− u
3
+ u
a
2
=
4.44062u
40
− 5.31684u
39
+ ··· + 13.7767u + 7.07457
−1.46774u
40
+ 1.54335u
39
+ ··· − 6.10178u − 1.46782
a
10
=
1.00290u
40
− 1.84189u
39
+ ··· + 5.12751u + 1.12239
−0.839015u
40
+ 1.67515u
39
+ ··· − 0.958478u − 0.838993
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
23238917313784
1716229488189
u
40
−
10481023528134
572076496063
u
39
+ ··· +
28692506559598
572076496063
u +
39302820789020
1716229488189
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
41
+ 8u
39
+ ··· + 687u + 229
c
2
u
41
+ 2u
40
+ ··· − 97u + 29
c
3
, c
10
u
41
+ 2u
40
+ ··· + 9u + 1
c
4
, c
8
u
41
+ 2u
40
+ ··· − u + 1
c
5
, c
7
u
41
+ 12u
40
+ ··· + 5u + 1
c
6
u
41
+ 4u
40
+ ··· + u + 1
c
9
u
41
− 7u
40
+ ··· + 6u − 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
41
+ 16y
40
+ ··· + 1271637y −52441
c
2
y
41
+ 48y
40
+ ··· − 7063y −841
c
3
, c
10
y
41
− 32y
40
+ ··· + 141y −1
c
4
, c
8
y
41
− 12y
40
+ ··· + 5y −1
c
5
, c
7
y
41
+ 36y
40
+ ··· + 5y −1
c
6
y
41
− 8y
40
+ ··· + 5y −1
c
9
y
41
+ 9y
40
+ ··· − 16y −4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.903206 + 0.396002I
a = 1.211920 + 0.667053I
b = −0.408078 + 0.420450I
−1.94008 − 1.34771I −7.61480 + 3.42502I
u = 0.903206 − 0.396002I
a = 1.211920 − 0.667053I
b = −0.408078 − 0.420450I
−1.94008 + 1.34771I −7.61480 − 3.42502I
u = −0.952455 + 0.222261I
a = −0.211328 + 0.932061I
b = −0.096162 + 0.914015I
−2.78156 + 3.84619I −6.97849 − 6.75687I
u = −0.952455 − 0.222261I
a = −0.211328 − 0.932061I
b = −0.096162 − 0.914015I
−2.78156 − 3.84619I −6.97849 + 6.75687I
u = −0.821825 + 0.760247I
a = 0.450451 − 0.306050I
b = 0.335334 − 0.324976I
3.49362 + 1.78935I 0.87448 − 4.34492I
u = −0.821825 − 0.760247I
a = 0.450451 + 0.306050I
b = 0.335334 + 0.324976I
3.49362 − 1.78935I 0.87448 + 4.34492I
u = 0.866850
a = 0.639506
b = −0.0835204
−1.43130 −6.87200
u = 0.798878 + 0.810698I
a = 0.377223 − 0.719870I
b = 0.265202 + 1.192170I
3.76974 + 2.25598I 1.50138 − 3.41744I
u = 0.798878 − 0.810698I
a = 0.377223 + 0.719870I
b = 0.265202 − 1.192170I
3.76974 − 2.25598I 1.50138 + 3.41744I
u = −1.102330 + 0.334587I
a = 0.338270 − 1.284530I
b = −1.253450 − 0.430907I
0.85372 + 8.63849I −1.84811 − 8.19635I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.102330 − 0.334587I
a = 0.338270 + 1.284530I
b = −1.253450 + 0.430907I
0.85372 − 8.63849I −1.84811 + 8.19635I
u = −0.058442 + 0.843545I
a = 1.57943 − 0.15589I
b = −1.284700 + 0.271028I
4.36449 − 4.62926I 4.68493 + 4.91932I
u = −0.058442 − 0.843545I
a = 1.57943 + 0.15589I
b = −1.284700 − 0.271028I
4.36449 + 4.62926I 4.68493 − 4.91932I
u = −0.883648 + 0.770325I
a = −3.62989 + 1.11022I
b = 1.134150 + 0.025263I
5.18095 + 2.90757I −15.8800 + 0.I
u = −0.883648 − 0.770325I
a = −3.62989 − 1.11022I
b = 1.134150 − 0.025263I
5.18095 − 2.90757I −15.8800 + 0.I
u = 0.760286 + 0.907829I
a = 1.73221 + 0.34788I
b = −1.45266 − 0.46932I
9.17940 + 7.97252I 3.27060 − 3.71618I
u = 0.760286 − 0.907829I
a = 1.73221 − 0.34788I
b = −1.45266 + 0.46932I
9.17940 − 7.97252I 3.27060 + 3.71618I
u = 0.866672 + 0.809557I
a = −1.56395 − 1.02120I
b = 1.60494 + 0.57469I
7.77351 − 0.87153I 6.95024 − 0.30904I
u = 0.866672 − 0.809557I
a = −1.56395 + 1.02120I
b = 1.60494 − 0.57469I
7.77351 + 0.87153I 6.95024 + 0.30904I
u = −0.929908 + 0.739442I
a = 0.037024 + 0.508793I
b = 0.197100 + 0.379729I
3.15987 + 3.90045I −0.31532 − 1.57295I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.929908 − 0.739442I
a = 0.037024 − 0.508793I
b = 0.197100 − 0.379729I
3.15987 − 3.90045I −0.31532 + 1.57295I
u = −0.744475 + 0.942248I
a = 1.62214 − 0.18187I
b = −1.310630 + 0.070763I
8.52020 + 0.56768I 6.81847 − 0.32338I
u = −0.744475 − 0.942248I
a = 1.62214 + 0.18187I
b = −1.310630 − 0.070763I
8.52020 − 0.56768I 6.81847 + 0.32338I
u = −0.739440 + 0.286658I
a = 0.11659 + 1.68574I
b = 1.073000 + 0.545709I
1.56831 + 2.54987I 1.57226 − 7.77175I
u = −0.739440 − 0.286658I
a = 0.11659 − 1.68574I
b = 1.073000 − 0.545709I
1.56831 − 2.54987I 1.57226 + 7.77175I
u = 0.913744 + 0.795988I
a = −2.19807 − 1.07996I
b = 1.55855 − 0.65970I
7.62795 − 5.14257I 6.43416 + 5.95767I
u = 0.913744 − 0.795988I
a = −2.19807 + 1.07996I
b = 1.55855 + 0.65970I
7.62795 + 5.14257I 6.43416 − 5.95767I
u = 1.191150 + 0.227807I
a = −0.076211 + 0.412771I
b = −1.116560 − 0.153454I
0.067811 + 0.953358I 2.02910 − 7.42558I
u = 1.191150 − 0.227807I
a = −0.076211 − 0.412771I
b = −1.116560 + 0.153454I
0.067811 − 0.953358I 2.02910 + 7.42558I
u = 0.960612 + 0.767884I
a = −1.113770 + 0.253317I
b = 0.163245 − 1.244780I
3.27367 − 8.18385I 0. + 8.35233I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.960612 − 0.767884I
a = −1.113770 − 0.253317I
b = 0.163245 + 1.244780I
3.27367 + 8.18385I 0. − 8.35233I
u = 0.748657 + 0.093307I
a = 0.47886 − 4.75552I
b = 0.938830 − 0.044980I
0.431185 − 0.284475I 13.1815 − 15.1990I
u = 0.748657 − 0.093307I
a = 0.47886 + 4.75552I
b = 0.938830 + 0.044980I
0.431185 + 0.284475I 13.1815 + 15.1990I
u = 1.022750 + 0.797628I
a = 1.90383 + 1.42431I
b = −1.44670 + 0.51890I
8.3544 − 14.2736I 0. + 8.33140I
u = 1.022750 − 0.797628I
a = 1.90383 − 1.42431I
b = −1.44670 − 0.51890I
8.3544 + 14.2736I 0. − 8.33140I
u = −1.045230 + 0.812235I
a = 1.41367 − 1.18781I
b = −1.277530 − 0.155370I
7.57848 + 5.87702I 0. − 5.65225I
u = −1.045230 − 0.812235I
a = 1.41367 + 1.18781I
b = −1.277530 + 0.155370I
7.57848 − 5.87702I 0. + 5.65225I
u = 0.039502 + 0.458105I
a = 0.880104 + 0.384951I
b = 0.142705 − 0.550416I
0.05414 − 1.50218I 0.18723 + 4.24532I
u = 0.039502 − 0.458105I
a = 0.880104 − 0.384951I
b = 0.142705 + 0.550416I
0.05414 + 1.50218I 0.18723 − 4.24532I
u = −0.361128 + 0.264161I
a = −0.168271 + 0.663616I
b = 1.275180 − 0.127224I
2.56290 − 0.10225I 4.38337 − 2.22967I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.361128 − 0.264161I
a = −0.168271 − 0.663616I
b = 1.275180 + 0.127224I
2.56290 + 0.10225I 4.38337 + 2.22967I
9
II. I
u
2
= hb − 1, a − 1, u − 1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
1
a
5
=
1
1
a
6
=
0
1
a
1
=
1
1
a
9
=
−1
0
a
3
=
2
1
a
7
=
1
1
a
2
=
1
0
a
10
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
u − 1
c
5
, c
8
, c
10
u + 1
c
9
u
11
(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
7
, c
8
, c
10
y −1
c
9
y
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 1.00000
0 0
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u
41
+ 8u
39
+ ··· + 687u + 229)
c
2
(u − 1)(u
41
+ 2u
40
+ ··· − 97u + 29)
c
3
(u − 1)(u
41
+ 2u
40
+ ··· + 9u + 1)
c
4
(u − 1)(u
41
+ 2u
40
+ ··· − u + 1)
c
5
(u + 1)(u
41
+ 12u
40
+ ··· + 5u + 1)
c
6
(u − 1)(u
41
+ 4u
40
+ ··· + u + 1)
c
7
(u − 1)(u
41
+ 12u
40
+ ··· + 5u + 1)
c
8
(u + 1)(u
41
+ 2u
40
+ ··· − u + 1)
c
9
u(u
41
− 7u
40
+ ··· + 6u − 2)
c
10
(u + 1)(u
41
+ 2u
40
+ ··· + 9u + 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
41
+ 16y
40
+ ··· + 1271637y −52441)
c
2
(y −1)(y
41
+ 48y
40
+ ··· − 7063y −841)
c
3
, c
10
(y −1)(y
41
− 32y
40
+ ··· + 141y −1)
c
4
, c
8
(y −1)(y
41
− 12y
40
+ ··· + 5y −1)
c
5
, c
7
(y −1)(y
41
+ 36y
40
+ ··· + 5y −1)
c
6
(y −1)(y
41
− 8y
40
+ ··· + 5y −1)
c
9
y(y
41
+ 9y
40
+ ··· − 16y −4)
15
