10
164
(K10n
38
)
A knot diagram
1
Linearized knot diagam
6 8 7 8 3 1 10 6 3 4
Solving Sequence
4,7 1,3
6 5 10 8 2 9
c
3
c
6
c
5
c
10
c
7
c
2
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u, −138u
11
+ 105u
10
+ ··· + 142a − 155, u
12
+ u
9
+ 6u
8
+ u
7
− u
6
+ 2u
5
+ 5u
4
− 2u
3
+ 2u + 1i
I
u
2
= h22976741298u
15
− 77906464811u
14
+ ··· + 11233228513b + 21004036137,
− 2983129u
15
+ 13995185u
14
+ ··· + 26682253a − 65290273, u
16
− 3u
15
+ ··· + 4u + 1i
I
u
3
= hb + u, 3u
5
+ u
4
+ 2u
3
− 3u
2
+ 2a + 6u + 1, u
6
+ u
4
− u
3
+ 3u
2
− u + 1i
* 3 irreducible components of dim
C
= 0, with total 34 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
= hb + u, −138u
11
+ 105u
10
+ · · · + 142a − 155, u
12
+ u
9
+ · · · + 2u + 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
1
=
0.971831u
11
− 0.739437u
10
+ ··· + 3.60563u + 1.09155
−u
a
3
=
1
−u
2
a
6
=
1.44718u
11
− 0.0739437u
10
+ ··· + 1.01056u + 3.60915
−0.464789u
11
+ 0.0492958u
10
+ ··· + 0.492958u − 0.739437
a
5
=
0.978873u
11
+ 0.0704225u
10
+ ··· + 0.204225u + 2.94366
−0.0669014u
11
− 0.193662u
10
+ ··· + 0.313380u − 0.595070
a
10
=
0.971831u
11
− 0.739437u
10
+ ··· + 2.60563u + 1.09155
−u
a
8
=
0.517606u
11
+ 0.0246479u
10
+ ··· − 0.00352113u + 2.13028
−0.464789u
11
+ 0.0492958u
10
+ ··· + 0.492958u − 0.739437
a
2
=
0.0422535u
11
+ 0.359155u
10
+ ··· − 0.408451u − 0.387324
0.468310u
11
− 0.144366u
10
+ ··· + 0.806338u + 0.665493
a
9
=
0.507042u
11
− 0.690141u
10
+ ··· + 3.09859u + 0.352113
−0.169014u
11
+ 0.0633803u
10
+ ··· − 1.36620u + 0.0492958
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
72
71
u
11
+
169
71
u
10
−
19
71
u
9
−
56
71
u
8
−
340
71
u
7
+
943
71
u
6
+
100
71
u
5
−
288
71
u
4
−
394
71
u
3
+
984
71
u
2
−
369
71
u−
192
71
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
12
+ 8u
11
+ ··· + 96u + 16
c
2
, c
8
u
12
− u
11
+ ··· − 2u + 1
c
3
, c
10
u
12
− u
9
+ 6u
8
− u
7
− u
6
− 2u
5
+ 5u
4
+ 2u
3
− 2u + 1
c
4
, c
9
u
12
− u
11
+ ··· + 4u
2
+ 2
c
5
u
12
− 9u
11
+ ··· − 20u + 16
c
7
u
12
+ 8u
11
+ ··· + 22u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
12
+ 6y
11
+ ··· − 640y + 256
c
2
, c
8
y
12
+ 17y
11
+ ··· + 6y + 1
c
3
, c
10
y
12
+ 12y
10
+ ··· − 4y + 1
c
4
, c
9
y
12
+ 5y
11
+ ··· + 16y + 4
c
5
y
12
− 11y
11
+ ··· + 80y + 256
c
7
y
12
+ 2y
11
+ ··· + 84y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.433167 + 0.820343I
a = 1.77573 − 0.27759I
b = −0.433167 − 0.820343I
−2.63922 − 4.58392I 1.89423 + 6.22117I
u = 0.433167 − 0.820343I
a = 1.77573 + 0.27759I
b = −0.433167 + 0.820343I
−2.63922 + 4.58392I 1.89423 − 6.22117I
u = −0.894529 + 0.606911I
a = −0.948486 − 0.965514I
b = 0.894529 − 0.606911I
−7.98844 + 5.04592I −3.50212 − 4.93530I
u = −0.894529 − 0.606911I
a = −0.948486 + 0.965514I
b = 0.894529 + 0.606911I
−7.98844 − 5.04592I −3.50212 + 4.93530I
u = 0.727666 + 0.459131I
a = 0.686537 − 0.236758I
b = −0.727666 − 0.459131I
−1.29616 − 0.86105I −4.70470 + 1.78151I
u = 0.727666 − 0.459131I
a = 0.686537 + 0.236758I
b = −0.727666 + 0.459131I
−1.29616 + 0.86105I −4.70470 − 1.78151I
u = −0.925706 + 1.050550I
a = −0.840738 + 0.491457I
b = 0.925706 − 1.050550I
3.38867 + 4.08003I −1.46265 − 0.78652I
u = −0.925706 − 1.050550I
a = −0.840738 − 0.491457I
b = 0.925706 + 1.050550I
3.38867 − 4.08003I −1.46265 + 0.78652I
u = −0.444254 + 0.260304I
a = −1.11367 + 2.11911I
b = 0.444254 − 0.260304I
1.58084 + 1.46904I 1.29817 − 5.01402I
u = −0.444254 − 0.260304I
a = −1.11367 − 2.11911I
b = 0.444254 + 0.260304I
1.58084 − 1.46904I 1.29817 + 5.01402I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.10366 + 1.16882I
a = 0.940626 + 0.241992I
b = −1.10366 − 1.16882I
−4.56023 − 12.50670I −0.52291 + 6.78913I
u = 1.10366 − 1.16882I
a = 0.940626 − 0.241992I
b = −1.10366 + 1.16882I
−4.56023 + 12.50670I −0.52291 − 6.78913I
6
II.
I
u
2
= h2.30×10
10
u
15
−7.79×10
10
u
14
+· · ·+1.12×10
10
b+2.10×10
10
, −2.98×
10
6
u
15
+1.40×10
7
u
14
+· · · +2.67×10
7
a−6.53×10
7
, u
16
−3u
15
+· · · +4u +1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
1
=
0.111802u
15
− 0.524513u
14
+ ··· − 0.695051u + 2.44696
−2.04543u
15
+ 6.93536u
14
+ ··· − 10.9045u − 1.86981
a
3
=
1
−u
2
a
6
=
0.888198u
15
− 2.47549u
14
+ ··· + 12.6951u + 1.55304
1.45142u
15
− 4.13944u
14
+ ··· + 13.6179u + 4.66725
a
5
=
2.23464u
15
− 6.47872u
14
+ ··· + 24.6683u + 6.03118
1.68993u
15
− 4.95565u
14
+ ··· + 15.1088u + 4.70335
a
10
=
−1.93362u
15
+ 6.41085u
14
+ ··· − 11.5996u + 0.577142
−2.04543u
15
+ 6.93536u
14
+ ··· − 10.9045u − 1.86981
a
8
=
2.50110u
15
− 6.52849u
14
+ ··· + 28.7759u + 8.29535
0.161484u
15
+ 0.0864436u
14
+ ··· + 4.46296u + 2.07506
a
2
=
−0.888198u
15
+ 2.47549u
14
+ ··· − 12.6951u − 1.55304
−1.34644u
15
+ 4.00323u
14
+ ··· − 11.9733u − 4.47814
a
9
=
0.0729519u
15
− 0.441341u
14
+ ··· − 0.188785u + 3.05693
−2.21877u
15
+ 7.30458u
14
+ ··· − 12.2278u − 2.70227
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
62712632292
11233228513
u
15
+
222551627880
11233228513
u
14
+ ··· −
386527091664
11233228513
u −
36877222054
11233228513
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
2
− u + 1)
8
c
2
, c
8
u
16
+ u
15
+ ··· − 48u + 19
c
3
, c
10
u
16
+ 3u
15
+ ··· − 4u + 1
c
4
, c
9
u
16
+ u
15
+ ··· − 6u + 1
c
5
(u
4
+ 3u
3
+ u
2
− 2u + 1)
4
c
7
(u
4
− u
3
+ u
2
+ 1)
4
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
2
+ y + 1)
8
c
2
, c
8
y
16
+ 15y
15
+ ··· + 2332y + 361
c
3
, c
10
y
16
+ 3y
15
+ ··· + 8y + 1
c
4
, c
9
y
16
+ 7y
15
+ ··· + 134y + 1
c
5
(y
4
− 7y
3
+ 15y
2
− 2y + 1)
4
c
7
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
4
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.051690 + 0.235939I
a = −0.276759 + 0.885546I
b = 0.44895 − 1.60911I
−5.14581 + 0.61478I −3.82674 + 1.44464I
u = 1.051690 − 0.235939I
a = −0.276759 − 0.885546I
b = 0.44895 + 1.60911I
−5.14581 − 0.61478I −3.82674 − 1.44464I
u = −0.804589 + 0.808792I
a = 0.847270 − 0.224662I
b = −0.88699 + 1.31736I
1.85594 + 5.19385I −0.17326 − 6.02890I
u = −0.804589 − 0.808792I
a = 0.847270 + 0.224662I
b = −0.88699 − 1.31736I
1.85594 − 5.19385I −0.17326 + 6.02890I
u = −0.321200 + 0.647019I
a = −0.766065 + 1.153070I
b = 0.160429 + 0.464095I
1.85594 + 1.13408I −0.173262 + 0.899303I
u = −0.321200 − 0.647019I
a = −0.766065 − 1.153070I
b = 0.160429 − 0.464095I
1.85594 − 1.13408I −0.173262 − 0.899303I
u = −0.160429 + 0.464095I
a = 1.99954 + 0.38616I
b = 0.321200 + 0.647019I
1.85594 − 1.13408I −0.173262 − 0.899303I
u = −0.160429 − 0.464095I
a = 1.99954 − 0.38616I
b = 0.321200 − 0.647019I
1.85594 + 1.13408I −0.173262 + 0.899303I
u = −0.311042 + 0.310121I
a = 2.19827 − 0.59252I
b = −1.60753 − 1.13440I
−5.14581 + 3.44499I −3.82674 − 8.37284I
u = −0.311042 − 0.310121I
a = 2.19827 + 0.59252I
b = −1.60753 + 1.13440I
−5.14581 − 3.44499I −3.82674 + 8.37284I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.88699 + 1.31736I
a = −0.628172 − 0.043405I
b = 0.804589 + 0.808792I
1.85594 − 5.19385I −0.17326 + 6.02890I
u = 0.88699 − 1.31736I
a = −0.628172 + 0.043405I
b = 0.804589 − 0.808792I
1.85594 + 5.19385I −0.17326 − 6.02890I
u = −0.44895 + 1.60911I
a = 0.579766 + 0.148974I
b = −1.051690 − 0.235939I
−5.14581 + 0.61478I −3.82674 + 1.44464I
u = −0.44895 − 1.60911I
a = 0.579766 − 0.148974I
b = −1.051690 + 0.235939I
−5.14581 − 0.61478I −3.82674 − 1.44464I
u = 1.60753 + 1.13440I
a = 0.046151 + 0.506163I
b = 0.311042 − 0.310121I
−5.14581 + 3.44499I −3.82674 − 8.37284I
u = 1.60753 − 1.13440I
a = 0.046151 − 0.506163I
b = 0.311042 + 0.310121I
−5.14581 − 3.44499I −3.82674 + 8.37284I
11
III.
I
u
3
= hb + u, 3u
5
+ u
4
+ 2u
3
− 3u
2
+ 2a + 6u + 1, u
6
+ u
4
− u
3
+ 3u
2
− u + 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
1
=
−
3
2
u
5
−
1
2
u
4
+ ··· − 3u −
1
2
−u
a
3
=
1
−u
2
a
6
=
−u
4
− u
2
+ u − 3
−
1
2
u
5
−
1
2
u
4
− u
3
+
1
2
u
2
−
1
2
a
5
=
−
1
2
u
5
−
3
2
u
4
− u
3
−
1
2
u
2
−
5
2
−u
4
− u
3
− u
2
− 1
a
10
=
−
3
2
u
5
−
1
2
u
4
+ ··· − 4u −
1
2
−u
a
8
=
−u
5
− 2u
4
− u
3
− u − 4
−
1
2
u
5
−
1
2
u
4
+
1
2
u
2
−
1
2
a
2
=
1
2
u
5
−
3
2
u
4
−
3
2
u
2
+ 3u −
7
2
1
2
u
5
+
1
2
u
4
+ ··· + u −
1
2
a
9
=
−2u
5
− u
4
− u
3
+ 2u
2
− 4u − 1
−
1
2
u
5
−
1
2
u
4
+ ··· − u −
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 7u
5
+ 5u
4
+ 6u
3
− 3u
2
+ 11u + 10
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
− u
5
+ 3u
4
− u
3
+ 3u
2
+ 2
c
2
, c
8
u
6
− u
5
+ 2u
4
− 2u
2
+ u + 1
c
3
, c
10
u
6
+ u
4
− u
3
+ 3u
2
− u + 1
c
4
, c
9
u
6
− u
5
+ 2u
4
+ 2u
3
− u
2
+ 2u + 2
c
5
u
6
+ 4u
5
+ 6u
4
+ 8u
3
+ 10u
2
+ 4u + 1
c
6
u
6
+ u
5
+ 3u
4
+ u
3
+ 3u
2
+ 2
c
7
u
6
+ 3u
5
+ 5u
4
+ 3u
3
+ u
2
+ 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
6
+ 5y
5
+ 13y
4
+ 21y
3
+ 21y
2
+ 12y + 4
c
2
, c
8
y
6
+ 3y
5
− 4y
3
+ 8y
2
− 5y + 1
c
3
, c
10
y
6
+ 2y
5
+ 7y
4
+ 7y
3
+ 9y
2
+ 5y + 1
c
4
, c
9
y
6
+ 3y
5
+ 6y
4
+ y
2
− 8y + 4
c
5
y
6
− 4y
5
− 8y
4
+ 26y
3
+ 48y
2
+ 4y + 1
c
7
y
6
+ y
5
+ 9y
4
+ 3y
3
+ 11y
2
+ 2y + 1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.747107 + 0.813589I
a = 0.239424 + 0.758194I
b = −0.747107 − 0.813589I
−4.92982 + 2.38212I −1.44137 − 0.69060I
u = 0.747107 − 0.813589I
a = 0.239424 − 0.758194I
b = −0.747107 + 0.813589I
−4.92982 − 2.38212I −1.44137 + 0.69060I
u = 0.125253 + 0.619808I
a = −1.46927 − 1.44270I
b = −0.125253 − 0.619808I
2.50509 − 1.44331I 12.78155 + 4.91052I
u = 0.125253 − 0.619808I
a = −1.46927 + 1.44270I
b = −0.125253 + 0.619808I
2.50509 + 1.44331I 12.78155 − 4.91052I
u = −0.87236 + 1.13524I
a = −0.770152 + 0.391132I
b = 0.87236 − 1.13524I
4.06966 + 4.74338I 5.65982 − 6.07362I
u = −0.87236 − 1.13524I
a = −0.770152 − 0.391132I
b = 0.87236 + 1.13524I
4.06966 − 4.74338I 5.65982 + 6.07362I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
8
)(u
6
− u
5
+ ··· + 3u
2
+ 2)(u
12
+ 8u
11
+ ··· + 96u + 16)
c
2
, c
8
(u
6
− u
5
+ 2u
4
− 2u
2
+ u + 1)(u
12
− u
11
+ ··· − 2u + 1)
· (u
16
+ u
15
+ ··· − 48u + 19)
c
3
, c
10
(u
6
+ u
4
− u
3
+ 3u
2
− u + 1)
· (u
12
− u
9
+ 6u
8
− u
7
− u
6
− 2u
5
+ 5u
4
+ 2u
3
− 2u + 1)
· (u
16
+ 3u
15
+ ··· − 4u + 1)
c
4
, c
9
(u
6
− u
5
+ 2u
4
+ 2u
3
− u
2
+ 2u + 2)(u
12
− u
11
+ ··· + 4u
2
+ 2)
· (u
16
+ u
15
+ ··· − 6u + 1)
c
5
(u
4
+ 3u
3
+ u
2
− 2u + 1)
4
(u
6
+ 4u
5
+ 6u
4
+ 8u
3
+ 10u
2
+ 4u + 1)
· (u
12
− 9u
11
+ ··· − 20u + 16)
c
6
((u
2
− u + 1)
8
)(u
6
+ u
5
+ ··· + 3u
2
+ 2)(u
12
+ 8u
11
+ ··· + 96u + 16)
c
7
(u
4
− u
3
+ u
2
+ 1)
4
(u
6
+ 3u
5
+ 5u
4
+ 3u
3
+ u
2
+ 1)
· (u
12
+ 8u
11
+ ··· + 22u + 4)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
2
+ y + 1)
8
(y
6
+ 5y
5
+ 13y
4
+ 21y
3
+ 21y
2
+ 12y + 4)
· (y
12
+ 6y
11
+ ··· − 640y + 256)
c
2
, c
8
(y
6
+ 3y
5
− 4y
3
+ 8y
2
− 5y + 1)(y
12
+ 17y
11
+ ··· + 6y + 1)
· (y
16
+ 15y
15
+ ··· + 2332y + 361)
c
3
, c
10
(y
6
+ 2y
5
+ ··· + 5y + 1)(y
12
+ 12y
10
+ ··· − 4y + 1)
· (y
16
+ 3y
15
+ ··· + 8y + 1)
c
4
, c
9
(y
6
+ 3y
5
+ 6y
4
+ y
2
− 8y + 4)(y
12
+ 5y
11
+ ··· + 16y + 4)
· (y
16
+ 7y
15
+ ··· + 134y + 1)
c
5
((y
4
− 7y
3
+ 15y
2
− 2y + 1)
4
)(y
6
− 4y
5
+ ··· + 4y + 1)
· (y
12
− 11y
11
+ ··· + 80y + 256)
c
7
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
4
(y
6
+ y
5
+ 9y
4
+ 3y
3
+ 11y
2
+ 2y + 1)
· (y
12
+ 2y
11
+ ··· + 84y + 16)
17
