11n
152
(K11n
152
)
A knot diagram
1
Linearized knot diagam
4 8 1 2 10 11 3 6 1 8 9
Solving Sequence
2,4 5,10
6 1 3 9 8 11 7
c
4
c
5
c
1
c
3
c
9
c
8
c
11
c
6
c
2
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
2
+ b − 2u, u
2
+ a − 2u + 1, 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
= hb + 1, u
4
− u
3
− 2u
2
+ a + u + 2, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
I
u
3
= hb + 1, a
5
+ 4a
4
+ 4a
3
− a
2
− 2a + 1, u + 1i
I
u
4
= hb + 1, −17u
9
+ 33u
8
− 84u
7
+ 13u
6
− 54u
5
− 142u
4
+ 20u
3
− 335u
2
+ 16a + 71u − 145,
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
= hu
2
+ b − 2u, u
2
+ a − 2u + 1, u
11
− 5u
10
+ · · · + 2u + 1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
10
=
−u
2
+ 2u − 1
−u
2
+ 2u
a
6
=
−u
6
+ 4u
5
− 5u
4
+ 4u
2
− 2u + 1
−u
6
+ 4u
5
− 4u
4
− 2u
3
+ 5u
2
a
1
=
u
u
a
3
=
−u
2
+ 1
−u
2
a
9
=
2u − 1
2u
a
8
=
−u
9
+ 4u
8
− 4u
7
− 6u
6
+ 13u
5
− 12u
3
+ 2u
2
+ 5u
−u
10
+ 3u
9
− u
8
− 8u
7
+ 8u
6
+ 7u
5
− 11u
4
− 4u
3
+ 5u
2
+ 3u + 1
a
11
=
−2u
2
+ 2u
−2u
2
+ u
a
7
=
2u
9
− 6u
8
+ 2u
7
+ 13u
6
− 12u
5
− 9u
4
+ 10u
3
+ 2u
2
− 2u + 1
2u
9
− 5u
8
+ 12u
6
− 6u
5
− 10u
4
+ 4u
3
+ 4u
2
a
7
=
2u
9
− 6u
8
+ 2u
7
+ 13u
6
− 12u
5
− 9u
4
+ 10u
3
+ 2u
2
− 2u + 1
2u
9
− 5u
8
+ 12u
6
− 6u
5
− 10u
4
+ 4u
3
+ 4u
2
(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
9
, c
11
u
11
− 5u
10
+ ··· + 2u + 1
c
2
, c
7
, c
10
u
11
+ u
10
+ ··· + 2u + 1
c
5
u
11
+ u
10
+ ··· − 33u
2
− 27
c
6
u
11
− u
10
+ 3u
8
+ 12u
7
+ 10u
6
− 6u
5
− 33u
4
− 31u
3
− 33u
2
− 10u − 11
c
8
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
9
, c
11
y
11
− 9y
10
+ ··· − 10y −1
c
2
, c
7
, c
10
y
11
− 9y
10
+ ··· − 2y −1
c
5
y
11
+ 15y
10
+ ··· − 1782y −729
c
6
y
11
− y
10
+ ··· − 626y −121
c
8
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 = −4.30835
b = −3.30835
−3.78211 32.5960
u = −0.832306 + 0.202239I
a = −3.31644 + 0.74113I
b = −2.31644 + 0.74113I
−2.76312 + 1.08944I −13.75530 + 1.30535I
u = −0.832306 − 0.202239I
a = −3.31644 − 0.74113I
b = −2.31644 − 0.74113I
−2.76312 − 1.08944I −13.75530 − 1.30535I
u = 1.263210 + 0.139301I
a = −0.0498765 − 0.0733316I
b = 0.950123 − 0.073332I
−8.16883 − 4.71969I −15.8344 + 7.6612I
u = 1.263210 − 0.139301I
a = −0.0498765 + 0.0733316I
b = 0.950123 + 0.073332I
−8.16883 + 4.71969I −15.8344 − 7.6612I
u = 1.31469 + 0.95832I
a = 0.819354 − 0.603155I
b = 1.81935 − 0.60315I
7.84139 − 5.06071I −4.48302 + 2.40182I
u = 1.31469 − 0.95832I
a = 0.819354 + 0.603155I
b = 1.81935 + 0.60315I
7.84139 + 5.06071I −4.48302 − 2.40182I
u = −0.113634 + 0.293281I
a = −1.154170 + 0.653215I
b = −0.154166 + 0.653215I
0.003691 + 1.266700I −0.27668 − 5.30833I
u = −0.113634 − 0.293281I
a = −1.154170 − 0.653215I
b = −0.154166 − 0.653215I
0.003691 − 1.266700I −0.27668 + 5.30833I
u = 1.40586 + 1.00997I
a = 0.855309 − 0.819815I
b = 1.85531 − 0.81981I
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 = 0.855309 + 0.819815I
b = 1.85531 + 0.81981I
7.4453 + 12.4339I −4.94880 − 5.95992I
6
II. I
u
2
= hb + 1, u
4
− u
3
− 2u
2
+ a + u + 2, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
10
=
−u
4
+ u
3
+ 2u
2
− u − 2
−1
a
6
=
u
4
− u
3
− 3u
2
+ 3u + 2
u + 1
a
1
=
u
u
a
3
=
−u
2
+ 1
−u
2
a
9
=
−u
4
+ u
3
+ 2u
2
− 2u − 2
−u − 1
a
8
=
u
3
− 2u
u
4
+ u
3
− u
2
− 2u − 1
a
11
=
−u
4
+ u
3
+ 2u
2
− u − 2
−1
a
7
=
1
u
2
a
7
=
1
u
2
(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
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
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1
c
9
(u − 1)
5
c
10
u
5
c
11
(u + 1)
5
8
(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
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
c
9
, c
11
(y −1)
5
c
10
y
5
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.21774
a = −1.82120
b = −1.00000
−4.04602 −15.9650
u = −0.309916 + 0.549911I
a = −1.77780 − 1.38013I
b = −1.00000
−1.97403 + 1.53058I −3.57269 − 4.45807I
u = −0.309916 − 0.549911I
a = −1.77780 + 1.38013I
b = −1.00000
−1.97403 − 1.53058I −3.57269 + 4.45807I
u = 1.41878 + 0.21917I
a = −0.311598 − 0.106340I
b = −1.00000
−7.51750 − 4.40083I −3.44484 + 1.78781I
u = 1.41878 − 0.21917I
a = −0.311598 + 0.106340I
b = −1.00000
−7.51750 + 4.40083I −3.44484 − 1.78781I
10
III. I
u
3
= hb + 1, a
5
+ 4a
4
+ 4a
3
− a
2
− 2a + 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
4
=
1
0
a
5
=
1
1
a
10
=
a
−1
a
6
=
−a
2
− a + 1
a + 2
a
1
=
−1
−1
a
3
=
0
−1
a
9
=
−1
−a − 2
a
8
=
0
a
4
+ 5a
3
+ 8a
2
+ 3a − 2
a
11
=
a
a
2
+ 3a + 1
a
7
=
0
a
4
+ 5a
3
+ 8a
2
+ 3a − 2
a
7
=
0
a
4
+ 5a
3
+ 8a
2
+ 3a − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3a
4
− 5a
3
+ 5a
2
+ 7a − 7
11
(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
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
8
u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1
c
10
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
11
u
5
− u
4
− 2u
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 −1)
5
c
2
, c
7
y
5
c
5
, c
9
, c
11
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
6
, c
10
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
8
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 = −1.30992 + 0.54991I
b = −1.00000
−1.97403 + 1.53058I −3.57269 − 4.45807I
u = −1.00000
a = −1.30992 − 0.54991I
b = −1.00000
−1.97403 − 1.53058I −3.57269 + 4.45807I
u = −1.00000
a = 0.418784 + 0.219165I
b = −1.00000
−7.51750 − 4.40083I −3.44484 + 1.78781I
u = −1.00000
a = 0.418784 − 0.219165I
b = −1.00000
−7.51750 + 4.40083I −3.44484 − 1.78781I
u = −1.00000
a = −2.21774
b = −1.00000
−4.04602 −15.9650
14
IV. I
u
4
= hb + 1, −17u
9
+ 33u
8
+ · · · + 16a − 145, u
10
− 2u
9
+ · · · + 8u − 1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
10
=
1.06250u
9
− 2.06250u
8
+ ··· − 4.43750u + 9.06250
−1
a
6
=
−1.31250u
9
+ 2.43750u
8
+ ··· + 4.93750u − 9.43750
0.0625000u
9
− 0.0625000u
8
+ ··· + 0.562500u + 1.06250
a
1
=
u
u
a
3
=
−u
2
+ 1
−u
2
a
9
=
u
9
− 2u
8
+ 5u
7
− u
6
+ 3u
5
+ 8u
4
− 2u
3
+ 19u
2
− 5u + 9
−0.0625000u
9
+ 0.0625000u
8
+ ··· − 0.562500u − 1.06250
a
8
=
1
4
u
9
−
1
2
u
8
+ ··· −
11
4
u + 3
1
4
u
7
−
1
4
u
6
+ ··· −
5
4
u −
1
4
a
11
=
1.18750u
9
− 2.18750u
8
+ ··· − 1.31250u + 11.1875
−0.312500u
9
+ 0.437500u
8
+ ··· + 0.937500u − 1.43750
a
7
=
−
1
4
u
9
−
1
2
u
8
+ ··· −
1
4
u − 3
−
3
4
u
8
+
1
4
u
7
+ ··· −
3
4
u +
1
2
a
7
=
−
1
4
u
9
−
1
2
u
8
+ ··· −
1
4
u − 3
−
3
4
u
8
+
1
4
u
7
+ ··· −
3
4
u +
1
2
(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
9
, c
11
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
10
u
10
+ u
9
+ ··· + 160u + 32
c
5
u
10
+ 2u
9
+ ··· − 100u − 43
c
6
u
10
− 10u
8
+ 43u
6
+ 17u
5
− 35u
4
+ 46u
3
+ 64u
2
− 38u − 29
c
8
(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
9
, c
11
y
10
+ 6y
9
+ ··· − 52y + 1
c
2
, c
7
, c
10
y
10
− 21y
9
+ ··· − 9728y + 1024
c
5
y
10
+ 20y
9
+ ··· − 13440y + 1849
c
6
y
10
− 20y
9
+ ··· − 5156y + 841
c
8
(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 = −0.406775 − 0.098778I
b = −1.00000
0.17487 + 2.21397I −2.88087 − 4.04855I
u = −0.394402 − 1.113210I
a = −0.406775 + 0.098778I
b = −1.00000
0.17487 − 2.21397I −2.88087 + 4.04855I
u = 0.124008 + 0.699342I
a = 0.640226 − 0.273116I
b = −1.00000
0.17487 − 2.21397I −2.88087 + 4.04855I
u = 0.124008 − 0.699342I
a = 0.640226 + 0.273116I
b = −1.00000
0.17487 + 2.21397I −2.88087 − 4.04855I
u = −1.30598
a = −0.898398
b = −1.00000
−2.52712 −3.66490
u = 0.93349 + 1.31744I
a = −0.565488 + 1.008900I
b = −1.00000
9.31336 − 3.33174I −3.28666 + 2.53508I
u = 0.93349 − 1.31744I
a = −0.565488 − 1.008900I
b = −1.00000
9.31336 + 3.33174I −3.28666 − 2.53508I
u = 0.92355 + 1.51424I
a = −0.639912 + 0.836095I
b = −1.00000
9.31336 + 3.33174I −3.28666 − 2.53508I
u = 0.92355 − 1.51424I
a = −0.639912 − 0.836095I
b = −1.00000
9.31336 − 3.33174I −3.28666 + 2.53508I
u = 0.132691
a = 8.84230
b = −1.00000
−2.52712 −3.66490
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
9
(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
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
11
(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
− 2u
3
− u
2
+ u − 1)(u
5
+ u
4
− u
3
− 4u
2
− 3u − 1)
· (u
10
+ 2u
9
+ ··· − 100u − 43)(u
11
+ u
10
+ ··· − 33u
2
− 27)
c
6
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)(u
5
+ u
4
− u
3
− 4u
2
− 3u − 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
7
, c
10
u
5
(u
5
+ u
4
+ ··· + u + 1)(u
10
+ u
9
+ ··· + 160u + 32)
· (u
11
+ u
10
+ ··· + 2u + 1)
c
8
(u
5
− u
4
+ u
2
+ u − 1)
2
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
2
· (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
9
, c
11
((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
10
y
5
(y
5
+ 3y
4
+ ··· − y −1)(y
10
− 21y
9
+ ··· − 9728y + 1024)
· (y
11
− 9y
10
+ ··· − 2y −1)
c
5
(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
6
(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
8
(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
