11n
151
(K11n
151
)
A knot diagram
1
Linearized knot diagam
4 8 1 2 9 10 3 11 1 8 6
Solving Sequence
8,10
11
3,9
2 7 6 1 5 4
c
10
c
8
c
2
c
7
c
6
c
11
c
5
c
4
c
1
, c
3
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
9
+ 4u
8
− 5u
7
− 2u
6
+ 9u
5
− 4u
4
− 4u
3
+ 4u
2
+ b − u,
u
9
− 4u
8
+ 4u
7
+ 6u
6
− 13u
5
+ 12u
3
− 2u
2
+ a − 5u,
u
11
− 5u
10
+ 8u
9
+ 3u
8
− 22u
7
+ 14u
6
+ 18u
5
− 19u
4
− 7u
3
+ 7u
2
+ 2u + 1i
I
u
2
= h−u
4
+ u
3
+ u
2
+ b − 1, a, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
I
u
3
= ha
4
+ 2a
2
+ b + 2, a
5
+ a
4
+ 2a
3
+ a
2
+ a + 1, u + 1i
I
u
4
= h−u
7
+ u
6
− u
5
− 2u
4
+ u
3
+ 4b + 5u + 1,
− 9u
9
+ 17u
8
− 44u
7
+ 5u
6
− 38u
5
− 78u
4
− 4u
3
− 207u
2
+ 16a + 7u − 113,
u
10
− 2u
9
+ 5u
8
− u
7
+ 3u
6
+ 8u
5
− 2u
4
+ 19u
3
− 6u
2
+ 8u − 1i
* 4 irreducible components of dim
C
= 0, with total 31 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
9
+4u
8
+· · · + b − u, u
9
−4u
8
+· · · + a − 5u, u
11
−5u
10
+· · · + 2u + 1i
(i) Arc colorings
a
8
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
3
=
−u
9
+ 4u
8
− 4u
7
− 6u
6
+ 13u
5
− 12u
3
+ 2u
2
+ 5u
u
9
− 4u
8
+ 5u
7
+ 2u
6
− 9u
5
+ 4u
4
+ 4u
3
− 4u
2
+ u
a
9
=
−u
−u
3
+ u
a
2
=
−u
9
+ 4u
8
− 4u
7
− 6u
6
+ 13u
5
− 12u
3
+ 2u
2
+ 5u
−u
10
+ 5u
9
− 7u
8
− 4u
7
+ 16u
6
− 3u
5
− 13u
4
+ 2u
3
+ 3u
2
+ 3u + 1
a
7
=
u
2
− 1
u
4
− 2u
3
+ 2u
a
6
=
u
4
− 2u
3
+ u
2
+ 2u − 1
u
4
− 2u
3
+ 2u
a
1
=
−u
10
+ 4u
9
− 5u
8
− 4u
7
+ 14u
6
− 6u
5
− 11u
4
+ 8u
3
+ 3u
2
− 2u + 1
−u
10
+ 4u
9
− 4u
8
− 6u
7
+ 13u
6
− 12u
4
+ 2u
3
+ 5u
2
a
5
=
u
8
− 2u
7
− u
6
+ 6u
5
− u
4
− 6u
3
+ 2u
2
+ 2u − 1
u
10
− 2u
9
− 2u
8
+ 8u
7
− u
6
− 10u
5
+ 4u
4
+ 2u
3
− u
2
+ 2u
a
4
=
u
10
− 4u
9
+ 5u
8
+ 4u
7
− 14u
6
+ 6u
5
+ 11u
4
− 8u
3
− 3u
2
+ 2u − 1
u
10
− 3u
9
− u
8
+ 14u
7
− 12u
6
− 15u
5
+ 21u
4
+ 6u
3
− 11u
2
− u − 1
a
4
=
u
10
− 4u
9
+ 5u
8
+ 4u
7
− 14u
6
+ 6u
5
+ 11u
4
− 8u
3
− 3u
2
+ 2u − 1
u
10
− 3u
9
− u
8
+ 14u
7
− 12u
6
− 15u
5
+ 21u
4
+ 6u
3
− 11u
2
− u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
10
− 16u
9
+ 20u
8
+ 8u
7
− 24u
6
− 16u
5
+ 28u
4
+ 32u
3
− 28u
2
− 24u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
8
, c
10
u
11
− 5u
10
+ ··· + 2u + 1
c
2
, c
7
, c
9
u
11
+ u
10
+ ··· + 2u + 1
c
5
u
11
− u
10
+ 3u
8
+ 12u
7
+ 10u
6
− 6u
5
− 33u
4
− 31u
3
− 33u
2
− 10u − 11
c
6
u
11
+ u
10
+ ··· − 33u
2
− 27
c
11
u
11
− u
10
+ u
8
+ 8u
7
− 12u
6
+ 8u
5
+ 3u
4
+ 3u
3
− 3u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
8
, c
10
y
11
− 9y
10
+ ··· − 10y −1
c
2
, c
7
, c
9
y
11
− 9y
10
+ ··· − 2y −1
c
5
y
11
− y
10
+ ··· − 626y −121
c
6
y
11
+ 15y
10
+ ··· − 1782y −729
c
11
y
11
− y
10
+ ··· + 6y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.07566
a = −0.382088
b = −6.69648
−3.78211 32.5960
u = −0.832306 + 0.202239I
a = −0.362795 + 0.658644I
b = −1.76349 − 0.21001I
−2.76312 + 1.08944I −13.75530 + 1.30535I
u = −0.832306 − 0.202239I
a = −0.362795 − 0.658644I
b = −1.76349 + 0.21001I
−2.76312 − 1.08944I −13.75530 − 1.30535I
u = 1.263210 + 0.139301I
a = 0.158505 − 0.711489I
b = 0.009586 + 0.293616I
−8.16883 − 4.71969I −15.8344 + 7.6612I
u = 1.263210 − 0.139301I
a = 0.158505 + 0.711489I
b = 0.009586 − 0.293616I
−8.16883 + 4.71969I −15.8344 − 7.6612I
u = 1.31469 + 0.95832I
a = −0.606321 + 1.088860I
b = −0.10260 − 1.75202I
7.84139 − 5.06071I −4.48302 + 2.40182I
u = 1.31469 − 0.95832I
a = −0.606321 − 1.088860I
b = −0.10260 + 1.75202I
7.84139 + 5.06071I −4.48302 − 2.40182I
u = −0.113634 + 0.293281I
a = −1.09164 + 1.49222I
b = 0.322788 + 0.550650I
0.003691 + 1.266700I −0.27668 − 5.30833I
u = −0.113634 − 0.293281I
a = −1.09164 − 1.49222I
b = 0.322788 − 0.550650I
0.003691 − 1.266700I −0.27668 + 5.30833I
u = 1.40586 + 1.00997I
a = 0.593293 − 1.135200I
b = 0.38195 + 1.94651I
7.4453 − 12.4339I −4.94880 + 5.95992I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.40586 − 1.00997I
a = 0.593293 + 1.135200I
b = 0.38195 − 1.94651I
7.4453 + 12.4339I −4.94880 − 5.95992I
6
II. I
u
2
= h−u
4
+ u
3
+ u
2
+ b − 1, a, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
8
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
3
=
0
u
4
− u
3
− u
2
+ 1
a
9
=
−u
−u
3
+ u
a
2
=
0
u
4
− u
3
− u
2
+ 1
a
7
=
0
u
a
6
=
u
u
a
1
=
−u
4
+ u
2
+ 1
−u
4
+ 2u
2
a
5
=
u
4
− u
2
− 1
u
4
− 2u
2
a
4
=
u
4
− u
2
− 1
2u
4
− u
3
− 3u
2
+ 1
a
4
=
u
4
− u
2
− 1
2u
4
− u
3
− 3u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
4
+ 7u
3
+ 2u
2
− 6u − 7
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
5
c
2
, c
7
u
5
c
3
, c
4
(u + 1)
5
c
5
, c
9
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
6
, c
8
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
10
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
11
u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
(y −1)
5
c
2
, c
7
y
5
c
5
, c
9
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
6
, c
8
, c
10
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
11
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.21774
a = 0
b = 3.52181
−4.04602 −15.9650
u = −0.309916 + 0.549911I
a = 0
b = 0.881366 + 0.489365I
−1.97403 + 1.53058I −3.57269 − 4.45807I
u = −0.309916 − 0.549911I
a = 0
b = 0.881366 − 0.489365I
−1.97403 − 1.53058I −3.57269 + 4.45807I
u = 1.41878 + 0.21917I
a = 0
b = −0.142272 + 0.509071I
−7.51750 − 4.40083I −3.44484 + 1.78781I
u = 1.41878 − 0.21917I
a = 0
b = −0.142272 − 0.509071I
−7.51750 + 4.40083I −3.44484 − 1.78781I
10
III. I
u
3
= ha
4
+ 2a
2
+ b + 2, a
5
+ a
4
+ 2a
3
+ a
2
+ a + 1, u + 1i
(i) Arc colorings
a
8
=
0
−1
a
10
=
1
0
a
11
=
1
1
a
3
=
a
−a
4
− 2a
2
− 2
a
9
=
1
0
a
2
=
a
−a
4
− 2a
2
+ a − 2
a
7
=
a
2
a
4
+ a
2
− a
a
6
=
a
4
+ 2a
2
− a
a
4
+ a
2
− a
a
1
=
−a
4
0
a
5
=
a
2
a
4
+ a
2
− a
a
4
=
−a
4
−a
4
− 2a
2
− 2
a
4
=
−a
4
−a
4
− 2a
2
− 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4a
4
− 3a
3
− 11a
2
− 2a − 11
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
2
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
3
, c
4
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
5
, c
6
u
5
− u
4
− u
3
+ 4u
2
− 3u + 1
c
7
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
8
(u − 1)
5
c
9
u
5
c
10
(u + 1)
5
c
11
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1
c
2
, c
7
y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1
c
5
, c
6
y
5
− 3y
4
+ 3y
3
− 8y
2
+ y − 1
c
8
, c
10
(y − 1)
5
c
9
y
5
c
11
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0.339110 + 0.822375I
b = −0.881366 − 0.489365I
−1.97403 + 1.53058I −3.57269 − 4.45807I
u = −1.00000
a = 0.339110 − 0.822375I
b = −0.881366 + 0.489365I
−1.97403 − 1.53058I −3.57269 + 4.45807I
u = −1.00000
a = −0.766826
b = −3.52181
−4.04602 −15.9650
u = −1.00000
a = −0. 455697 + 1.200150I
b = 0.142272 − 0.509071I
−7.51750 − 4.40083I −3.44484 + 1.78781I
u = −1.00000
a = −0. 455697 − 1.200150I
b = 0.142272 + 0.509071I
−7.51750 + 4.40083I −3.44484 − 1.78781I
14
IV. I
u
4
= h−u
7
+ u
6
− u
5
− 2u
4
+ u
3
+ 4b + 5u + 1, −9u
9
+ 17u
8
+ · · · +
16a − 113, u
10
− 2u
9
+ · · · + 8u − 1i
(i) Arc colorings
a
8
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
3
=
0.562500u
9
− 1.06250u
8
+ ··· − 0.437500u + 7.06250
1
4
u
7
−
1
4
u
6
+ ··· −
5
4
u −
1
4
a
9
=
−u
−u
3
+ u
a
2
=
0.562500u
9
− 1.06250u
8
+ ··· − 0.437500u + 7.06250
0.0625000u
9
− 0.0625000u
8
+ ··· − 1.18750u − 0.187500
a
7
=
−
3
2
u
9
+
19
8
u
8
+ ··· + u −
53
8
0.312500u
9
− 0.437500u
8
+ ··· + 0.0625000u + 0.437500
a
6
=
−1.18750u
9
+ 1.93750u
8
+ ··· + 1.06250u − 6.18750
0.312500u
9
− 0.437500u
8
+ ··· + 0.0625000u + 0.437500
a
1
=
−
1
4
u
9
+
1
2
u
8
+ ··· +
3
2
u −
13
4
1
4
u
8
−
1
2
u
7
+ ··· + u +
1
4
a
5
=
−u
9
+
11
8
u
8
+ ··· −
5
2
u −
45
8
−0.437500u
9
+ 0.312500u
8
+ ··· + 5.31250u − 0.312500
a
4
=
1
4
u
9
−
1
2
u
8
+ ··· −
3
2
u +
13
4
−
1
4
u
8
+
1
4
u
7
+ ··· +
5
4
u
2
+
1
4
u
a
4
=
1
4
u
9
−
1
2
u
8
+ ··· −
3
2
u +
13
4
−
1
4
u
8
+
1
4
u
7
+ ··· +
5
4
u
2
+
1
4
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
1
4
u
9
+
7
8
u
8
−
9
4
u
7
+ 2u
6
−
3
8
u
5
−
23
8
u
4
+
33
8
u
3
−
35
8
u
2
+
31
4
u −
37
8
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
8
, c
10
u
10
− 2u
9
+ 5u
8
− u
7
+ 3u
6
+ 8u
5
− 2u
4
+ 19u
3
− 6u
2
+ 8u − 1
c
2
, c
7
, c
9
u
10
+ u
9
+ ··· + 160u + 32
c
5
u
10
− 10u
8
+ 43u
6
+ 17u
5
− 35u
4
+ 46u
3
+ 64u
2
− 38u − 29
c
6
u
10
+ 2u
9
+ ··· − 100u − 43
c
11
(u
5
− u
4
+ u
2
+ u − 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
8
, c
10
y
10
+ 6y
9
+ ··· − 52y + 1
c
2
, c
7
, c
9
y
10
− 21y
9
+ ··· − 9728y + 1024
c
5
y
10
− 20y
9
+ ··· − 5156y + 841
c
6
y
10
+ 20y
9
+ ··· − 13440y + 1849
c
11
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y − 1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.394402 + 1 .113210I
a = −0. 721708 − 0.484512I
b = −0.233677 + 0.885557I
0.17487 + 2.21397I −2.88087 − 4.04855I
u = −0.394402 − 1.113210I
a = −0. 721708 + 0.484512I
b = −0.233677 − 0.885557I
0.17487 − 2.21397I −2.88087 + 4.04855I
u = 0.124008 + 0.699342I
a = 1.91026 + 1.28243I
b = −0.233677 − 0.885557I
0.17487 − 2.21397I −2.88087 + 4.04855I
u = 0.124008 − 0.699342I
a = 1.91026 − 1.28243I
b = −0.233677 + 0.885557I
0.17487 + 2.21397I −2.88087 − 4.04855I
u = −1.30598
a = −0.276456
b = −0.416284
−2.52712 −3.66490
u = 0.93349 + 1.31744I
a = −0.80555 + 1.36977I
b = −0.05818 − 1.69128I
9.31336 − 3.33174I −3.28666 + 2.53508I
u = 0.93349 − 1.31744I
a = −0.80555 − 1.36977I
b = −0.05818 + 1.69128I
9.31336 + 3.33174I −3.28666 − 2.53508I
u = 0.92355 + 1.51424I
a = 0.638018 − 1.084890I
b = −0.05818 + 1.69128I
9.31336 + 3.33174I −3.28666 − 2.53508I
u = 0.92355 − 1.51424I
a = 0.638018 + 1.084890I
b = −0.05818 − 1.69128I
9.31336 − 3.33174I −3.28666 + 2.53508I
u = 0.132691
a = 7.23443
b = −0.416284
−2.52712 −3.66490
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
8
(u − 1)
5
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
· (u
10
− 2u
9
+ 5u
8
− u
7
+ 3u
6
+ 8u
5
− 2u
4
+ 19u
3
− 6u
2
+ 8u − 1)
· (u
11
− 5u
10
+ ··· + 2u + 1)
c
2
, c
9
u
5
(u
5
− u
4
+ ··· + u − 1)(u
10
+ u
9
+ ··· + 160u + 32)
· (u
11
+ u
10
+ ··· + 2u + 1)
c
3
, c
4
, c
10
(u + 1)
5
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
· (u
10
− 2u
9
+ 5u
8
− u
7
+ 3u
6
+ 8u
5
− 2u
4
+ 19u
3
− 6u
2
+ 8u − 1)
· (u
11
− 5u
10
+ ··· + 2u + 1)
c
5
(u
5
− u
4
− u
3
+ 4u
2
− 3u + 1)(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
· (u
10
− 10u
8
+ 43u
6
+ 17u
5
− 35u
4
+ 46u
3
+ 64u
2
− 38u − 29)
· (u
11
− u
10
+ 3u
8
+ 12u
7
+ 10u
6
− 6u
5
− 33u
4
− 31u
3
− 33u
2
− 10u − 11)
c
6
(u
5
− u
4
− u
3
+ 4u
2
− 3u + 1)(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
· (u
10
+ 2u
9
+ ··· − 100u − 43)(u
11
+ u
10
+ ··· − 33u
2
− 27)
c
7
u
5
(u
5
+ u
4
+ ··· + u + 1)(u
10
+ u
9
+ ··· + 160u + 32)
· (u
11
+ u
10
+ ··· + 2u + 1)
c
11
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)(u
5
− u
4
+ u
2
+ u − 1)
2
· (u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
· (u
11
− u
10
+ u
8
+ 8u
7
− 12u
6
+ 8u
5
+ 3u
4
+ 3u
3
− 3u
2
+ 1)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
8
, c
10
((y − 1)
5
)(y
5
− 5y
4
+ ··· − y − 1)(y
10
+ 6y
9
+ ··· − 52y + 1)
· (y
11
− 9y
10
+ ··· − 10y − 1)
c
2
, c
7
, c
9
y
5
(y
5
+ 3y
4
+ ··· − y − 1)(y
10
− 21y
9
+ ··· − 9728y + 1024)
· (y
11
− 9y
10
+ ··· − 2y − 1)
c
5
(y
5
− 3y
4
+ 3y
3
− 8y
2
+ y − 1)(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
· (y
10
− 20y
9
+ ··· − 5156y + 841)(y
11
− y
10
+ ··· − 626y − 121)
c
6
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)(y
5
− 3y
4
+ 3y
3
− 8y
2
+ y − 1)
· (y
10
+ 20y
9
+ ··· − 13440y + 1849)
· (y
11
+ 15y
10
+ ··· − 1782y − 729)
c
11
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y − 1)
2
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1)
2
· (y
11
− y
10
+ ··· + 6y − 1)
20
