11n
40
(K11n
40
)
A knot diagram
1
Linearized knot diagam
4 1 6 2 11 4 10 6 5 8 9
Solving Sequence
1,4
2
5,9
10 11 6 7 3 8
c
1
c
4
c
9
c
11
c
5
c
6
c
3
c
8
c
2
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h8.07539 × 10
36
u
46
+ 6.00242 × 10
37
u
45
+ ··· + 8.54755 × 10
36
b + 1.34031 × 10
37
,
1.86673 × 10
35
u
46
− 1.48910 × 10
36
u
45
+ ··· + 4.27377 × 10
36
a + 1.07223 × 10
37
, u
47
+ 8u
46
+ ··· + 7u + 1i
I
u
2
= h−3a
5
+ 13a
4
− 7a
3
− 17a
2
+ 13b − 21a + 7, a
6
− 6a
5
+ 11a
4
− 4a
3
− a
2
− a + 1, u − 1i
I
u
3
= hb, a − 3u − 5, u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 55 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
= h8.08 × 10
36
u
46
+ 6.00 × 10
37
u
45
+ · · · + 8.55 × 10
36
b + 1.34 × 10
37
, 1.87 ×
10
35
u
46
−1.49×10
36
u
45
+· · ·+4.27×10
36
a+1.07×10
37
, u
47
+8u
46
+· · ·+7u+1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
9
=
−0.0436786u
46
+ 0.348428u
45
+ ··· − 27.9405u − 2.50886
−0.944761u
46
− 7.02239u
45
+ ··· − 8.76360u − 1.56807
a
10
=
0.183235u
46
+ 2.58132u
45
+ ··· − 20.0808u − 1.19485
−2.50742u
46
− 18.7010u
45
+ ··· − 19.7734u − 3.29966
a
11
=
−1.08058u
46
− 8.14247u
45
+ ··· − 9.68656u + 1.51066
−0.508743u
46
− 3.94324u
45
+ ··· − 9.53527u − 1.05799
a
6
=
0.377639u
46
+ 2.54848u
45
+ ··· − 12.1966u − 0.249306
0.653298u
46
+ 5.04572u
45
+ ··· + 6.43493u + 1.03094
a
7
=
0.377639u
46
+ 2.54848u
45
+ ··· − 12.1966u − 0.249306
1.15585u
46
+ 8.72441u
45
+ ··· + 9.36570u + 1.50357
a
3
=
−u
2
+ 1
u
2
a
8
=
−0.334510u
46
− 1.43080u
45
+ ··· − 34.1017u − 1.22443
−1.47305u
46
− 11.2678u
45
+ ··· − 14.9627u − 2.38719
a
8
=
−0.334510u
46
− 1.43080u
45
+ ··· − 34.1017u − 1.22443
−1.47305u
46
− 11.2678u
45
+ ··· − 14.9627u − 2.38719
(ii) Obstruction class = −1
(iii) Cusp Shapes = −7.05583u
46
− 61.7041u
45
+ ··· + 99.8008u + 14.7497
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
47
− 8u
46
+ ··· + 7u − 1
c
2
u
47
+ 18u
46
+ ··· − 3u + 1
c
3
, c
6
u
47
− 2u
46
+ ··· − 64u − 64
c
5
u
47
− 3u
46
+ ··· + 2u − 1
c
7
, c
10
u
47
+ 4u
46
+ ··· − 11u − 1
c
8
u
47
+ 3u
46
+ ··· + 698u + 191
c
9
u
47
− u
46
+ ··· − 3568u − 5873
c
11
u
47
− 8u
46
+ ··· + 48u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
47
− 18y
46
+ ··· − 3y −1
c
2
y
47
+ 30y
46
+ ··· − 1935y −1
c
3
, c
6
y
47
+ 36y
46
+ ··· − 61440y −4096
c
5
y
47
+ y
46
+ ··· + 8y −1
c
7
, c
10
y
47
− 38y
46
+ ··· + 407y −1
c
8
y
47
− 59y
46
+ ··· + 1536176y −36481
c
9
y
47
− 19y
46
+ ··· + 74984424y −34492129
c
11
y
47
+ 12y
46
+ ··· + 1080y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.033480 + 0.093725I
a = −4.34666 − 0.37811I
b = −0.484814 + 0.321200I
0.321927 − 0.588102I −6.8283 − 18.9142I
u = 1.033480 − 0.093725I
a = −4.34666 + 0.37811I
b = −0.484814 − 0.321200I
0.321927 + 0.588102I −6.8283 + 18.9142I
u = −0.757104 + 0.786690I
a = −0.082397 + 1.043700I
b = −0.149310 − 0.794646I
3.72033 + 1.52573I −5.00000 − 4.80548I
u = −0.757104 − 0.786690I
a = −0.082397 − 1.043700I
b = −0.149310 + 0.794646I
3.72033 − 1.52573I −5.00000 + 4.80548I
u = 0.764975 + 0.478588I
a = −0.05345 − 1.98335I
b = 0.667437 + 1.036640I
−1.05831 − 3.36011I −6.88945 + 7.26716I
u = 0.764975 − 0.478588I
a = −0.05345 + 1.98335I
b = 0.667437 − 1.036640I
−1.05831 + 3.36011I −6.88945 − 7.26716I
u = −0.698652 + 0.895191I
a = −0.993201 − 0.828921I
b = 0.82225 + 1.68360I
4.30107 − 2.55894I 0
u = −0.698652 − 0.895191I
a = −0.993201 + 0.828921I
b = 0.82225 − 1.68360I
4.30107 + 2.55894I 0
u = 1.202260 + 0.035924I
a = −0.067077 − 0.771324I
b = 0.578274 − 0.672866I
−2.41059 + 1.46028I 0
u = 1.202260 − 0.035924I
a = −0.067077 + 0.771324I
b = 0.578274 + 0.672866I
−2.41059 − 1.46028I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.898968 + 0.805280I
a = 1.48193 − 0.16674I
b = 0.682111 + 0.148627I
5.31855 + 3.02042I 0
u = −0.898968 − 0.805280I
a = 1.48193 + 0.16674I
b = 0.682111 − 0.148627I
5.31855 − 3.02042I 0
u = −0.778806 + 0.103648I
a = −1.004530 + 0.086037I
b = −1.151800 + 0.485072I
−1.17157 + 5.91398I 2.20637 − 8.69493I
u = −0.778806 − 0.103648I
a = −1.004530 − 0.086037I
b = −1.151800 − 0.485072I
−1.17157 − 5.91398I 2.20637 + 8.69493I
u = 0.855886 + 0.862796I
a = −0.143884 + 1.345810I
b = −0.720152 − 1.125900I
3.93862 − 7.66972I 0
u = 0.855886 − 0.862796I
a = −0.143884 − 1.345810I
b = −0.720152 + 1.125900I
3.93862 + 7.66972I 0
u = −0.998921 + 0.724997I
a = −0.824145 − 0.951628I
b = −0.454183 + 0.696516I
2.96538 + 4.21460I 0
u = −0.998921 − 0.724997I
a = −0.824145 + 0.951628I
b = −0.454183 − 0.696516I
2.96538 − 4.21460I 0
u = −0.861156 + 0.894994I
a = −0.33542 − 1.57495I
b = −1.77332 + 1.19079I
8.18563 + 0.96335I 0
u = −0.861156 − 0.894994I
a = −0.33542 + 1.57495I
b = −1.77332 − 1.19079I
8.18563 − 0.96335I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.690875 + 0.281167I
a = −1.63585 + 0.73751I
b = 0.155632 − 0.399985I
−0.875787 − 0.039510I −8.12380 − 0.07387I
u = 0.690875 − 0.281167I
a = −1.63585 − 0.73751I
b = 0.155632 + 0.399985I
−0.875787 + 0.039510I −8.12380 + 0.07387I
u = −0.565814 + 1.147400I
a = 0.586418 + 0.893805I
b = −0.92728 − 1.37308I
10.31450 − 8.43955I 0
u = −0.565814 − 1.147400I
a = 0.586418 − 0.893805I
b = −0.92728 + 1.37308I
10.31450 + 8.43955I 0
u = −0.969617 + 0.854773I
a = 0.945992 + 0.082576I
b = −1.47588 − 1.57575I
7.84802 + 5.50326I 0
u = −0.969617 − 0.854773I
a = 0.945992 − 0.082576I
b = −1.47588 + 1.57575I
7.84802 − 5.50326I 0
u = 0.687161
a = 11.6667
b = 0.145926
0.618242 −202.120
u = 0.989321 + 0.869498I
a = 0.710026 − 0.507400I
b = −0.247996 + 0.981313I
3.55960 + 1.25869I 0
u = 0.989321 − 0.869498I
a = 0.710026 + 0.507400I
b = −0.247996 − 0.981313I
3.55960 − 1.25869I 0
u = −0.526112 + 1.208050I
a = −0.180970 − 0.781461I
b = 0.148746 + 1.186400I
9.82610 + 0.01608I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.526112 − 1.208050I
a = −0.180970 + 0.781461I
b = 0.148746 − 1.186400I
9.82610 − 0.01608I 0
u = −1.065440 + 0.776679I
a = 0.79380 + 1.47518I
b = 1.23888 − 1.53370I
3.17396 + 8.77694I 0
u = −1.065440 − 0.776679I
a = 0.79380 − 1.47518I
b = 1.23888 + 1.53370I
3.17396 − 8.77694I 0
u = −0.626077 + 0.139965I
a = 0.942781 + 0.803760I
b = 1.225360 + 0.246198I
−2.68564 − 0.62982I −2.91172 − 0.97884I
u = −0.626077 − 0.139965I
a = 0.942781 − 0.803760I
b = 1.225360 − 0.246198I
−2.68564 + 0.62982I −2.91172 + 0.97884I
u = −1.21414 + 0.79886I
a = −0.80618 − 1.32555I
b = −1.15564 + 1.28689I
8.2500 + 15.4047I 0
u = −1.21414 − 0.79886I
a = −0.80618 + 1.32555I
b = −1.15564 − 1.28689I
8.2500 − 15.4047I 0
u = 0.434172 + 0.311062I
a = 0.51325 − 2.05221I
b = −0.968161 + 0.658442I
2.25397 − 1.36700I 1.30471 + 4.47621I
u = 0.434172 − 0.311062I
a = 0.51325 + 2.05221I
b = −0.968161 − 0.658442I
2.25397 + 1.36700I 1.30471 − 4.47621I
u = 0.525437
a = −1.52938
b = 0.222426
−0.954527 −10.1140
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.25537 + 0.82211I
a = 0.569872 + 0.706242I
b = 0.530975 − 1.012180I
7.53187 + 7.19737I 0
u = −1.25537 − 0.82211I
a = 0.569872 − 0.706242I
b = 0.530975 + 1.012180I
7.53187 − 7.19737I 0
u = 1.52770 + 0.06121I
a = −0.230584 − 0.148433I
b = −0.522294 − 0.982637I
1.89096 − 4.57089I 0
u = 1.52770 − 0.06121I
a = −0.230584 + 0.148433I
b = −0.522294 + 0.982637I
1.89096 + 4.57089I 0
u = −1.59066
a = −0.00631231
b = 0.284188
−7.32077 0
u = −0.093469 + 0.137034I
a = 1.09478 − 2.52730I
b = −0.345092 − 0.814513I
1.82947 + 1.07812I 2.48829 − 1.79959I
u = −0.093469 − 0.137034I
a = 1.09478 + 2.52730I
b = −0.345092 + 0.814513I
1.82947 − 1.07812I 2.48829 + 1.79959I
9
II. I
u
2
=
h−3a
5
+13a
4
−7a
3
−17a
2
+13b−21a+7, a
6
−6a
5
+11a
4
−4a
3
−a
2
−a+1, u−1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
9
=
a
3
13
a
5
− a
4
+ ··· +
21
13
a −
7
13
a
10
=
−
3
13
a
5
+ a
4
+ ··· −
8
13
a +
7
13
3
13
a
5
− a
4
+ ··· +
21
13
a −
7
13
a
11
=
−
5
13
a
5
+ 2a
4
+ ··· +
4
13
a +
16
13
−1.15385a
5
+ 6a
4
+ ··· + 0.923077a + 0.692308
a
6
=
0
−2.07692a
5
+ 11a
4
+ ··· + 2.46154a + 2.84615
a
7
=
0
−2.07692a
5
+ 11a
4
+ ··· + 2.46154a + 2.84615
a
3
=
0
1
a
8
=
a
−1.92308a
5
+ 10a
4
+ ··· + 0.538462a + 2.15385
a
8
=
a
−1.92308a
5
+ 10a
4
+ ··· + 0.538462a + 2.15385
(ii) Obstruction class = 1
(iii) Cusp Shapes =
28
13
a
5
− 11a
4
+
178
13
a
3
+
85
13
a
2
−
12
13
a −
204
13
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
6
c
2
, c
4
(u + 1)
6
c
3
, c
6
u
6
c
5
, c
8
u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1
c
7
, c
9
, c
11
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
10
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
6
y
6
c
5
, c
8
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
c
7
, c
9
, c
10
c
11
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.658836 + 0.177500I
b = 1.073950 + 0.558752I
−1.64493 − 5.69302I −11.70582 + 2.69056I
u = 1.00000
a = 0.658836 − 0.177500I
b = 1.073950 − 0.558752I
−1.64493 + 5.69302I −11.70582 − 2.69056I
u = 1.00000
a = −0.346225 + 0.393823I
b = −1.002190 + 0.295542I
−3.53554 + 0.92430I −13.12292 − 1.33143I
u = 1.00000
a = −0.346225 − 0.393823I
b = −1.002190 − 0.295542I
−3.53554 − 0.92430I −13.12292 + 1.33143I
u = 1.00000
a = 2.68739 + 0.76772I
b = 0.428243 − 0.664531I
0.245672 − 0.924305I −5.17126 + 7.13914I
u = 1.00000
a = 2.68739 − 0.76772I
b = 0.428243 + 0.664531I
0.245672 + 0.924305I −5.17126 − 7.13914I
13
III. I
u
3
= hb, a − 3u − 5, u
2
+ u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
−u + 1
a
5
=
−u
−u + 1
a
9
=
3u + 5
0
a
10
=
2u + 4
1
a
11
=
1
0
a
6
=
−1
−u + 1
a
7
=
−1
0
a
3
=
u
−u + 1
a
8
=
2u + 3
1
a
8
=
2u + 3
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 41
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
2
+ u − 1
c
2
, c
5
u
2
+ 3u + 1
c
4
, c
6
u
2
− u − 1
c
7
(u + 1)
2
c
8
, c
9
u
2
− 3u + 1
c
10
(u − 1)
2
c
11
u
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
y
2
− 3y + 1
c
2
, c
5
, c
8
c
9
y
2
− 7y + 1
c
7
, c
10
(y −1)
2
c
11
y
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 6.85410
b = 0
0.657974 41.0000
u = −1.61803
a = 0.145898
b = 0
−7.23771 41.0000
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
2
+ u − 1)(u
47
− 8u
46
+ ··· + 7u − 1)
c
2
((u + 1)
6
)(u
2
+ 3u + 1)(u
47
+ 18u
46
+ ··· − 3u + 1)
c
3
u
6
(u
2
+ u − 1)(u
47
− 2u
46
+ ··· − 64u − 64)
c
4
((u + 1)
6
)(u
2
− u − 1)(u
47
− 8u
46
+ ··· + 7u − 1)
c
5
(u
2
+ 3u + 1)(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
· (u
47
− 3u
46
+ ··· + 2u − 1)
c
6
u
6
(u
2
− u − 1)(u
47
− 2u
46
+ ··· − 64u − 64)
c
7
((u + 1)
2
)(u
6
− u
5
+ ··· − u + 1)(u
47
+ 4u
46
+ ··· − 11u − 1)
c
8
(u
2
− 3u + 1)(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
· (u
47
+ 3u
46
+ ··· + 698u + 191)
c
9
(u
2
− 3u + 1)(u
6
− u
5
+ ··· − u + 1)(u
47
− u
46
+ ··· − 3568u − 5873)
c
10
((u − 1)
2
)(u
6
+ u
5
+ ··· + u + 1)(u
47
+ 4u
46
+ ··· − 11u − 1)
c
11
u
2
(u
6
− u
5
+ ··· − u + 1)(u
47
− 8u
46
+ ··· + 48u + 4)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
6
)(y
2
− 3y + 1)(y
47
− 18y
46
+ ··· − 3y −1)
c
2
((y −1)
6
)(y
2
− 7y + 1)(y
47
+ 30y
46
+ ··· − 1935y −1)
c
3
, c
6
y
6
(y
2
− 3y + 1)(y
47
+ 36y
46
+ ··· − 61440y −4096)
c
5
(y
2
− 7y + 1)(y
6
+ y
5
+ ··· + 3y + 1)(y
47
+ y
46
+ ··· + 8y −1)
c
7
, c
10
(y −1)
2
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
· (y
47
− 38y
46
+ ··· + 407y −1)
c
8
(y
2
− 7y + 1)(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
· (y
47
− 59y
46
+ ··· + 1536176y −36481)
c
9
(y
2
− 7y + 1)(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
· (y
47
− 19y
46
+ ··· + 74984424y −34492129)
c
11
y
2
(y
6
− 3y
5
+ ··· − y + 1)(y
47
+ 12y
46
+ ··· + 1080y −16)
19
