11n
137
(K11n
137
)
A knot diagram
1
Linearized knot diagam
8 1 9 8 11 10 2 4 6 7 9
Solving Sequence
6,9
10 7 11
1,4
3 2 5 8
c
9
c
6
c
10
c
11
c
3
c
2
c
5
c
8
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
15
− u
14
+ 6u
13
+ 4u
12
− 15u
11
− 3u
10
+ 19u
9
− 7u
8
− 10u
7
+ 12u
6
− 2u
5
− 3u
4
+ 4u
3
− 2u
2
+ b + 1,
u
15
+ u
14
− 5u
13
− 4u
12
+ 9u
11
+ 4u
10
− 6u
9
+ 2u
8
− u
7
− 4u
6
+ 4u
5
− 4u
3
+ 2a + u − 1,
u
16
+ 3u
15
+ ··· − 3u − 2i
I
u
2
= h−4u
8
a + 6u
8
+ ··· − 3a + 4,
− 2u
8
a + 8u
6
a + 2u
7
+ 2u
5
a + u
6
− 9u
4
a − 7u
5
− 6u
3
a − 5u
4
+ u
2
a + 6u
3
+ a
2
+ 4au + 7u
2
− a + 2u − 2,
u
9
− u
8
− 4u
7
+ 3u
6
+ 5u
5
− u
4
− 2u
3
− 2u
2
+ u − 1i
I
u
3
= hu
5
− 2u
3
+ b + u, u
5
− 3u
3
− u
2
+ a + 2u + 1, u
6
− 3u
4
+ 2u
2
+ 1i
* 3 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
= h−u
15
−u
14
+· · ·+b+1, u
15
+u
14
+· · ·+2a−1, u
16
+3u
15
+· · ·−3u−2i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
7
=
u
−u
3
+ u
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
1
=
−u
4
+ u
2
+ 1
u
4
− 2u
2
a
4
=
−
1
2
u
15
−
1
2
u
14
+ ··· −
1
2
u +
1
2
u
15
+ u
14
+ ··· + 2u
2
− 1
a
3
=
−
3
2
u
15
−
3
2
u
14
+ ··· −
1
2
u +
3
2
u
15
+ u
14
+ ··· + 2u
2
− 1
a
2
=
1
2
u
15
+
1
2
u
14
+ ··· −
1
2
u +
1
2
−u
15
− u
14
+ ··· + u + 1
a
5
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
8
=
3
2
u
15
+
5
2
u
14
+ ··· −
5
2
u −
3
2
−u
15
− 2u
14
+ ··· − 2u
2
+ 1
a
8
=
3
2
u
15
+
5
2
u
14
+ ··· −
5
2
u −
3
2
−u
15
− 2u
14
+ ··· − 2u
2
+ 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
15
+ 6u
14
− 20u
13
− 22u
12
+ 42u
11
+ 12u
10
− 44u
9
+ 42u
8
+
4u
7
− 54u
6
+ 40u
5
− 6u
4
− 28u
3
+ 20u
2
− 12u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
u
16
+ 2u
14
+ ··· + 2u − 1
c
2
u
16
+ 4u
15
+ ··· − 14u
2
+ 1
c
5
u
16
+ 9u
15
+ ··· + 31u + 22
c
6
, c
9
, c
10
u
16
− 3u
15
+ ··· + 3u − 2
c
11
u
16
− 3u
15
+ ··· − 41u − 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
y
16
+ 4y
15
+ ··· − 14y
2
+ 1
c
2
y
16
+ 24y
15
+ ··· − 28y + 1
c
5
y
16
− 3y
15
+ ··· − 3557y + 484
c
6
, c
9
, c
10
y
16
− 15y
15
+ ··· − 21y + 4
c
11
y
16
+ 9y
15
+ ··· − 6561y + 64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.608375 + 0.583971I
a = 0.147239 − 0.217444I
b = −0.826528 + 0.979522I
3.36394 − 4.13872I 7.73528 + 1.97260I
u = 0.608375 − 0.583971I
a = 0.147239 + 0.217444I
b = −0.826528 − 0.979522I
3.36394 + 4.13872I 7.73528 − 1.97260I
u = 0.395219 + 0.742683I
a = −1.25592 + 1.19798I
b = 0.797243 + 1.086110I
2.59863 + 8.63192I 5.90792 − 7.27043I
u = 0.395219 − 0.742683I
a = −1.25592 − 1.19798I
b = 0.797243 − 1.086110I
2.59863 − 8.63192I 5.90792 + 7.27043I
u = 1.216880 + 0.292072I
a = 0.840694 − 0.714472I
b = −0.494247 − 0.784033I
0.99780 + 5.12268I 7.85223 − 7.82309I
u = 1.216880 − 0.292072I
a = 0.840694 + 0.714472I
b = −0.494247 + 0.784033I
0.99780 − 5.12268I 7.85223 + 7.82309I
u = 0.012792 + 0.713635I
a = 0.244689 − 1.197750I
b = 0.379775 − 0.677130I
−2.70658 − 1.45405I 3.73411 + 4.71917I
u = 0.012792 − 0.713635I
a = 0.244689 + 1.197750I
b = 0.379775 + 0.677130I
−2.70658 + 1.45405I 3.73411 − 4.71917I
u = −1.271500 + 0.260922I
a = −0.319754 − 0.233539I
b = −0.232716 − 0.644221I
1.24387 − 2.05073I 9.21244 − 1.11358I
u = −1.271500 − 0.260922I
a = −0.319754 + 0.233539I
b = −0.232716 + 0.644221I
1.24387 + 2.05073I 9.21244 + 1.11358I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.43214
a = −1.38066
b = 0.888414
6.54271 14.4520
u = −1.47068 + 0.28044I
a = 2.09438 + 0.18309I
b = −0.82217 + 1.15830I
8.6109 − 12.3641I 9.35094 + 7.14528I
u = −1.47068 − 0.28044I
a = 2.09438 − 0.18309I
b = −0.82217 − 1.15830I
8.6109 + 12.3641I 9.35094 − 7.14528I
u = −1.50706 + 0.17257I
a = −1.11020 − 1.03847I
b = 0.966111 + 0.941274I
10.25570 + 1.47993I 11.45831 − 1.74331I
u = −1.50706 − 0.17257I
a = −1.11020 + 1.03847I
b = 0.966111 − 0.941274I
10.25570 − 1.47993I 11.45831 + 1.74331I
u = −0.400197
a = 0.598381
b = −0.423356
0.656537 15.0460
6
II.
I
u
2
= h−4u
8
a+6u
8
+· · ·−3a+4, −2u
8
a+2u
7
+· · ·−a−2, u
9
−u
8
+· · ·+u−1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
7
=
u
−u
3
+ u
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
1
=
−u
4
+ u
2
+ 1
u
4
− 2u
2
a
4
=
a
4u
8
a − 6u
8
+ ··· + 3a − 4
a
3
=
−4u
8
a + 6u
8
+ ··· − 2a + 4
4u
8
a − 6u
8
+ ··· + 3a − 4
a
2
=
−3u
8
a + 6u
8
+ ··· − 2a + 5
5u
8
a − 8u
8
+ ··· + 4a − 6
a
5
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
8
=
6u
8
a − 9u
8
+ ··· + 4a − 7
−2u
8
a + 3u
8
+ ··· − 2a + 2
a
8
=
6u
8
a − 9u
8
+ ··· + 4a − 7
−2u
8
a + 3u
8
+ ··· − 2a + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
6
+ 12u
4
+ 4u
3
− 8u
2
− 8u + 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
u
18
− u
17
+ ··· − 8u + 5
c
2
u
18
+ 7u
17
+ ··· + 136u + 25
c
5
(u
9
− 3u
8
+ 2u
7
+ 5u
6
− u
5
− 13u
4
+ 10u
3
+ 2u
2
+ u − 3)
2
c
6
, c
9
, c
10
(u
9
+ u
8
− 4u
7
− 3u
6
+ 5u
5
+ u
4
− 2u
3
+ 2u
2
+ u + 1)
2
c
11
(u
9
− u
8
+ 6u
7
− 5u
6
+ 11u
5
− 7u
4
+ 6u
3
− 2u
2
+ u − 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
y
18
+ 7y
17
+ ··· + 136y + 25
c
2
y
18
+ 7y
17
+ ··· + 5004y + 625
c
5
(y
9
− 5y
8
+ 32y
7
− 87y
6
+ 185y
5
− 223y
4
+ 180y
3
− 62y
2
+ 13y −9)
2
c
6
, c
9
, c
10
(y
9
− 9y
8
+ 32y
7
− 55y
6
+ 45y
5
− 19y
4
+ 16y
3
− 10y
2
− 3y −1)
2
c
11
(y
9
+ 11y
8
+ 48y
7
+ 105y
6
+ 121y
5
+ 73y
4
+ 20y
3
− 6y
2
− 3y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.482242 + 0.666986I
a = 1.119660 + 0.834506I
b = −0.881705 + 0.851729I
3.77376 − 2.21388I 8.24115 + 3.04598I
u = −0.482242 + 0.666986I
a = −0.009091 − 0.470353I
b = 0.937576 + 0.708026I
3.77376 − 2.21388I 8.24115 + 3.04598I
u = −0.482242 − 0.666986I
a = 1.119660 − 0.834506I
b = −0.881705 − 0.851729I
3.77376 + 2.21388I 8.24115 − 3.04598I
u = −0.482242 − 0.666986I
a = −0.009091 + 0.470353I
b = 0.937576 − 0.708026I
3.77376 + 2.21388I 8.24115 − 3.04598I
u = 1.28056
a = 1.66854 + 0.09359I
b = −0.295309 + 1.123220I
−0.453072 5.66670
u = 1.28056
a = 1.66854 − 0.09359I
b = −0.295309 − 1.123220I
−0.453072 5.66670
u = −1.380230 + 0.162431I
a = −0.931046 + 0.163673I
b = 0.076831 − 1.264200I
1.87293 − 3.41073I 9.88238 + 4.39642I
u = −1.380230 + 0.162431I
a = 1.54746 − 0.88517I
b = −0.505863 + 0.476260I
1.87293 − 3.41073I 9.88238 + 4.39642I
u = −1.380230 − 0.162431I
a = −0.931046 − 0.163673I
b = 0.076831 + 1.264200I
1.87293 + 3.41073I 9.88238 − 4.39642I
u = −1.380230 − 0.162431I
a = 1.54746 + 0.88517I
b = −0.505863 − 0.476260I
1.87293 + 3.41073I 9.88238 − 4.39642I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.230908 + 0.456719I
a = −1.90677 − 0.85951I
b = 0.257033 + 0.703723I
−3.25448 + 1.10969I 4.55374 − 6.23947I
u = 0.230908 + 0.456719I
a = 0.96790 − 1.89385I
b = 0.033137 − 1.191070I
−3.25448 + 1.10969I 4.55374 − 6.23947I
u = 0.230908 − 0.456719I
a = −1.90677 + 0.85951I
b = 0.257033 − 0.703723I
−3.25448 − 1.10969I 4.55374 + 6.23947I
u = 0.230908 − 0.456719I
a = 0.96790 + 1.89385I
b = 0.033137 + 1.191070I
−3.25448 − 1.10969I 4.55374 + 6.23947I
u = 1.49128 + 0.23430I
a = 1.01299 − 1.10233I
b = −1.067290 + 0.668745I
10.17130 + 5.50049I 11.48937 − 2.97298I
u = 1.49128 + 0.23430I
a = −1.96964 − 0.01296I
b = 0.945590 + 0.965095I
10.17130 + 5.50049I 11.48937 − 2.97298I
u = 1.49128 − 0.23430I
a = 1.01299 + 1.10233I
b = −1.067290 − 0.668745I
10.17130 − 5.50049I 11.48937 + 2.97298I
u = 1.49128 − 0.23430I
a = −1.96964 + 0.01296I
b = 0.945590 − 0.965095I
10.17130 − 5.50049I 11.48937 + 2.97298I
11
III. I
u
3
= hu
5
− 2u
3
+ b + u, u
5
− 3u
3
− u
2
+ a + 2u + 1, u
6
− 3u
4
+ 2u
2
+ 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
7
=
u
−u
3
+ u
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
1
=
−u
4
+ u
2
+ 1
u
4
− 2u
2
a
4
=
−u
5
+ 3u
3
+ u
2
− 2u − 1
−u
5
+ 2u
3
− u
a
3
=
u
3
+ u
2
− u − 1
−u
5
+ 2u
3
− u
a
2
=
−u
4
+ u
3
+ 2u
2
− u
−u
5
+ u
4
+ 2u
3
− 2u
2
− u
a
5
=
−u
5
+ 2u
3
− u
0
a
8
=
−u
4
− u
3
+ 2u
2
+ 2u
−1
a
8
=
−u
4
− u
3
+ 2u
2
+ 2u
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
4
+ 8u
2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
(u
2
+ 1)
3
c
2
(u + 1)
6
c
5
u
6
+ u
4
+ 2u
2
+ 1
c
6
, c
9
, c
10
u
6
− 3u
4
+ 2u
2
+ 1
c
11
(u
3
− u
2
+ 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
(y + 1)
6
c
2
(y − 1)
6
c
5
(y
3
+ y
2
+ 2y + 1)
2
c
6
, c
9
, c
10
(y
3
− 3y
2
+ 2y + 1)
2
c
11
(y
3
− y
2
+ 2y − 1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.307140 + 0.215080I
a = 1.40722 + 0.43972I
b = − 1.000000I
−0.26574 + 2.82812I 3.50976 − 2.97945I
u = 1.307140 − 0.215080I
a = 1.40722 − 0.43972I
b = 1.000000I
−0.26574 − 2.82812I 3.50976 + 2.97945I
u = −1.307140 + 0.215080I
a = −0.082503 − 0.684841I
b = − 1.000000I
−0.26574 − 2.82812I 3.50976 + 2.97945I
u = −1.307140 − 0.215080I
a = −0.082503 + 0.684841I
b = 1.000000I
−0.26574 + 2.82812I 3.50976 − 2.97945I
u = 0.569840I
a = −1.32472 − 1.75488I
b = − 1.000000I
−4.40332 −3.01950
u = − 0.569840I
a = −1.32472 + 1.75488I
b = 1.000000I
−4.40332 −3.01950
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
((u
2
+ 1)
3
)(u
16
+ 2u
14
+ ··· + 2u − 1)(u
18
− u
17
+ ··· − 8u + 5)
c
2
((u + 1)
6
)(u
16
+ 4u
15
+ ··· − 14u
2
+ 1)(u
18
+ 7u
17
+ ··· + 136u + 25)
c
5
(u
6
+ u
4
+ 2u
2
+ 1)
· (u
9
− 3u
8
+ 2u
7
+ 5u
6
− u
5
− 13u
4
+ 10u
3
+ 2u
2
+ u − 3)
2
· (u
16
+ 9u
15
+ ··· + 31u + 22)
c
6
, c
9
, c
10
(u
6
− 3u
4
+ 2u
2
+ 1)
· (u
9
+ u
8
− 4u
7
− 3u
6
+ 5u
5
+ u
4
− 2u
3
+ 2u
2
+ u + 1)
2
· (u
16
− 3u
15
+ ··· + 3u − 2)
c
11
((u
3
− u
2
+ 1)
2
)(u
9
− u
8
+ ··· + u − 1)
2
· (u
16
− 3u
15
+ ··· − 41u − 8)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
((y + 1)
6
)(y
16
+ 4y
15
+ ··· − 14y
2
+ 1)(y
18
+ 7y
17
+ ··· + 136y + 25)
c
2
((y − 1)
6
)(y
16
+ 24y
15
+ ··· − 28y + 1)(y
18
+ 7y
17
+ ··· + 5004y + 625)
c
5
(y
3
+ y
2
+ 2y + 1)
2
· (y
9
− 5y
8
+ 32y
7
− 87y
6
+ 185y
5
− 223y
4
+ 180y
3
− 62y
2
+ 13y − 9)
2
· (y
16
− 3y
15
+ ··· − 3557y + 484)
c
6
, c
9
, c
10
(y
3
− 3y
2
+ 2y + 1)
2
· (y
9
− 9y
8
+ 32y
7
− 55y
6
+ 45y
5
− 19y
4
+ 16y
3
− 10y
2
− 3y − 1)
2
· (y
16
− 15y
15
+ ··· − 21y + 4)
c
11
(y
3
− y
2
+ 2y − 1)
2
· (y
9
+ 11y
8
+ 48y
7
+ 105y
6
+ 121y
5
+ 73y
4
+ 20y
3
− 6y
2
− 3y − 1)
2
· (y
16
+ 9y
15
+ ··· − 6561y + 64)
17
