11n
53
(K11n
53
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 9 10 4 1 11 6 7
Solving Sequence
7,10
6 11
1,4
2 3 9 5 8
c
6
c
10
c
11
c
1
c
2
c
9
c
5
c
8
c
3
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
22
+ 2u
21
+ ··· + b + 1, u
21
− u
20
+ ··· + 2u
3
+ a, u
23
− 2u
22
+ ··· − 2u + 1i
I
u
2
= hb, −u
2
+ a − u − 1, u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 28 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
22
+2u
21
+· · ·+b+1, u
21
−u
20
+· · ·+2u
3
+a, u
23
−2u
22
+· · ·−2u+1i
(i) Arc colorings
a
7
=
1
0
a
10
=
0
u
a
6
=
1
−u
2
a
11
=
−u
u
3
+ u
a
1
=
u
3
u
3
+ u
a
4
=
−u
21
+ u
20
+ ··· − u
4
− 2u
3
u
22
− 2u
21
+ ··· + 2u − 1
a
2
=
u
18
− u
17
+ ··· − u
2
+ u
−u
8
− 2u
6
− 2u
4
a
3
=
−u
21
+ u
20
+ ··· + 2u − 1
u
22
− 2u
21
+ ··· + u − 1
a
9
=
−u
3
u
5
+ u
3
+ u
a
5
=
−u
6
− u
4
+ 1
u
8
+ 2u
6
+ 2u
4
a
8
=
−u
11
− 2u
9
− 2u
7
− u
3
−u
11
− 3u
9
− 4u
7
− u
5
+ u
3
+ u
a
8
=
−u
11
− 2u
9
− 2u
7
− u
3
−u
11
− 3u
9
− 4u
7
− u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
22
−9u
21
+ 30u
20
−48u
19
+ 96u
18
−131u
17
+ 190u
16
−224u
15
+ 249u
14
−262u
13
+
237u
12
− 210u
11
+ 148u
10
− 93u
9
+ 53u
8
− 6u
7
− 14u
6
+ 25u
5
− 18u
4
− 10u
2
+ 9u − 5
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
23
− 6u
22
+ ··· + 4u − 1
c
2
u
23
+ 32u
22
+ ··· − 16u + 1
c
3
, c
7
u
23
− u
22
+ ··· + 32u − 32
c
5
, c
11
u
23
− 2u
22
+ ··· + 30u − 9
c
6
, c
10
u
23
+ 2u
22
+ ··· − 2u − 1
c
8
u
23
+ 24u
21
+ ··· + 2u − 1
c
9
u
23
− 12u
22
+ ··· − 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
23
− 32y
22
+ ··· − 16y −1
c
2
y
23
− 76y
22
+ ··· + 772y −1
c
3
, c
7
y
23
+ 33y
22
+ ··· + 3584y −1024
c
5
, c
11
y
23
− 12y
22
+ ··· − 162y −81
c
6
, c
10
y
23
+ 12y
22
+ ··· − 2y −1
c
8
y
23
+ 48y
22
+ ··· − 2y −1
c
9
y
23
+ 24y
21
+ ··· − 6y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.700458 + 0.794163I
a = −1.45877 + 0.07887I
b = −0.09416 − 2.13003I
−14.1192 + 2.6463I −3.65714 − 2.84707I
u = −0.700458 − 0.794163I
a = −1.45877 − 0.07887I
b = −0.09416 + 2.13003I
−14.1192 − 2.6463I −3.65714 + 2.84707I
u = 0.855800 + 0.265895I
a = −0.795468 − 0.188526I
b = −0.45428 − 1.93201I
−11.07970 + 5.13429I −2.96602 − 2.08249I
u = 0.855800 − 0.265895I
a = −0.795468 + 0.188526I
b = −0.45428 + 1.93201I
−11.07970 − 5.13429I −2.96602 + 2.08249I
u = 0.366516 + 1.072630I
a = 0.312367 − 0.791611I
b = −0.215406 − 0.997383I
1.62847 − 1.16584I 3.47504 + 0.42481I
u = 0.366516 − 1.072630I
a = 0.312367 + 0.791611I
b = −0.215406 + 0.997383I
1.62847 + 1.16584I 3.47504 − 0.42481I
u = −0.477094 + 1.041350I
a = 2.43046 − 1.25941I
b = 1.105500 + 0.395247I
−0.82985 + 3.19017I 0.92409 − 4.72756I
u = −0.477094 − 1.041350I
a = 2.43046 + 1.25941I
b = 1.105500 − 0.395247I
−0.82985 − 3.19017I 0.92409 + 4.72756I
u = 0.270789 + 0.755843I
a = 0.693663 + 0.260910I
b = 0.349805 − 0.349927I
0.390924 − 1.217980I 4.26810 + 5.45735I
u = 0.270789 − 0.755843I
a = 0.693663 − 0.260910I
b = 0.349805 + 0.349927I
0.390924 + 1.217980I 4.26810 − 5.45735I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.514060 + 1.104640I
a = −1.31154 + 0.63853I
b = 0.143374 + 1.329070I
0.55803 − 6.09572I 0.93537 + 6.19372I
u = 0.514060 − 1.104640I
a = −1.31154 − 0.63853I
b = 0.143374 − 1.329070I
0.55803 + 6.09572I 0.93537 − 6.19372I
u = −0.455035 + 1.182370I
a = −1.061190 + 0.524858I
b = −0.608111 − 0.025252I
5.15559 + 4.27437I 9.78577 − 3.28837I
u = −0.455035 − 1.182370I
a = −1.061190 − 0.524858I
b = −0.608111 + 0.025252I
5.15559 − 4.27437I 9.78577 + 3.28837I
u = 0.269522 + 1.238660I
a = 0.24803 + 2.58133I
b = 0.34057 + 1.76704I
−6.23110 + 1.56382I 1.72734 − 0.04821I
u = 0.269522 − 1.238660I
a = 0.24803 − 2.58133I
b = 0.34057 − 1.76704I
−6.23110 − 1.56382I 1.72734 + 0.04821I
u = −0.719779
a = 0.526584
b = 0.562756
1.81820 6.35980
u = 0.573488 + 1.178530I
a = 2.84278 − 1.00616I
b = 0.57374 − 1.89861I
−8.35091 − 10.39480I −0.03516 + 5.66347I
u = 0.573488 − 1.178530I
a = 2.84278 + 1.00616I
b = 0.57374 + 1.89861I
−8.35091 + 10.39480I −0.03516 − 5.66347I
u = −0.469209 + 0.502755I
a = −0.699918 − 0.973698I
b = −0.674614 + 0.656598I
−2.46377 + 0.78545I −4.30142 − 0.09221I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.469209 − 0.502755I
a = −0.699918 + 0.973698I
b = −0.674614 − 0.656598I
−2.46377 − 0.78545I −4.30142 + 0.09221I
u = 0.611511 + 0.287188I
a = 0.036277 − 0.858010I
b = −0.247796 + 1.071450I
−1.75612 + 1.64275I −3.33585 − 2.40342I
u = 0.611511 − 0.287188I
a = 0.036277 + 0.858010I
b = −0.247796 − 1.071450I
−1.75612 − 1.64275I −3.33585 + 2.40342I
7
II. I
u
2
= hb, −u
2
+ a − u − 1, u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
7
=
1
0
a
10
=
0
u
a
6
=
1
−u
2
a
11
=
−u
u
3
+ u
a
1
=
u
3
u
3
+ u
a
4
=
u
2
+ u + 1
0
a
2
=
u
3
+ u
2
+ u + 1
u
3
+ u
a
3
=
u
2
+ u + 1
0
a
9
=
−u
3
−u
4
− u
3
− u
2
− 1
a
5
=
−u
3
−u
3
− u
a
8
=
1
0
a
8
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
4
+ u
3
+ 2u
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
5
c
2
, c
4
(u + 1)
5
c
3
, c
7
u
5
c
5
, c
8
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
6
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
9
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
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
7
y
5
c
5
, c
8
, 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
9
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.339110 + 0.822375I
a = 0.77780 + 1.38013I
b = 0
−1.31583 − 1.53058I 0.02124 + 2.62456I
u = 0.339110 − 0.822375I
a = 0.77780 − 1.38013I
b = 0
−1.31583 + 1.53058I 0.02124 − 2.62456I
u = −0.766826
a = 0.821196
b = 0
0.756147 −2.67610
u = −0.455697 + 1.200150I
a = −0.688402 + 0.106340I
b = 0
4.22763 + 4.40083I 0.31681 − 3.97407I
u = −0.455697 − 1.200150I
a = −0.688402 − 0.106340I
b = 0
4.22763 − 4.40083I 0.31681 + 3.97407I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
23
− 6u
22
+ ··· + 4u − 1)
c
2
((u + 1)
5
)(u
23
+ 32u
22
+ ··· − 16u + 1)
c
3
, c
7
u
5
(u
23
− u
22
+ ··· + 32u − 32)
c
4
((u + 1)
5
)(u
23
− 6u
22
+ ··· + 4u − 1)
c
5
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
23
− 2u
22
+ ··· + 30u − 9)
c
6
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
23
+ 2u
22
+ ··· − 2u − 1)
c
8
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
23
+ 24u
21
+ ··· + 2u − 1)
c
9
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)(u
23
− 12u
22
+ ··· − 2u + 1)
c
10
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)(u
23
+ 2u
22
+ ··· − 2u − 1)
c
11
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
23
− 2u
22
+ ··· + 30u − 9)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
5
)(y
23
− 32y
22
+ ··· − 16y −1)
c
2
((y −1)
5
)(y
23
− 76y
22
+ ··· + 772y −1)
c
3
, c
7
y
5
(y
23
+ 33y
22
+ ··· + 3584y −1024)
c
5
, c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
23
− 12y
22
+ ··· − 162y −81)
c
6
, c
10
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
23
+ 12y
22
+ ··· − 2y −1)
c
8
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
23
+ 48y
22
+ ··· − 2y −1)
c
9
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)(y
23
+ 24y
21
+ ··· − 6y −1)
13
