11a
108
(K11a
108
)
A knot diagram
1
Linearized knot diagam
5 1 9 10 2 11 3 4 8 6 7
Solving Sequence
4,9 1,3
2 8 10 5 7 11 6
c
3
c
2
c
8
c
9
c
4
c
7
c
11
c
6
c
1
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−4u
40
+ 4u
39
+ ··· + 4b + 4, −2u
40
+ 4u
39
+ ··· + 4a − 2, u
41
− 2u
40
+ ··· − 4u + 2i
I
u
2
= h46u
4
a
2
+ 10u
4
a + ··· − 33a + 56,
− 2u
3
a
2
− u
4
a − 2a
2
u
2
+ 2u
3
a + 3u
4
+ a
3
− 2a
2
u + u
2
a + u
3
− 2a
2
+ au + 2u
2
+ 2a + 3u + 1,
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
I
u
3
= hu
3
+ b + u − 1, u
3
− 2u
2
+ 2a − 4, u
4
+ 2u
2
+ 2i
I
v
1
= ha, b + 1, v −1i
* 4 irreducible components of dim
C
= 0, with total 61 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−4u
40
+4u
39
+· · ·+4b+4, −2u
40
+4u
39
+· · ·+4a−2, u
41
−2u
40
+· · ·−4u+2i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
1
=
1
2
u
40
− u
39
+ ··· +
1
2
u +
1
2
u
40
− u
39
+ ··· +
1
2
u − 1
a
3
=
1
−u
2
a
2
=
−
1
4
u
33
− 2u
31
+ ··· −
1
2
u
3
+ 1
1
4
u
35
+
9
4
u
33
+ ··· −
3
2
u
2
−
1
2
u
a
8
=
u
u
a
10
=
u
3
u
3
+ u
a
5
=
−u
6
− u
4
+ 1
−u
6
− 2u
4
− u
2
a
7
=
−u
3
u
5
+ u
3
+ u
a
11
=
−
1
4
u
36
−
9
4
u
34
+ ··· −
1
2
u +
1
2
−
1
4
u
36
−
5
2
u
34
+ ··· −
1
2
u
2
− u
a
6
=
−
1
4
u
36
−
9
4
u
34
+ ··· −
1
2
u +
1
2
1
4
u
38
+
5
2
u
36
+ ··· + u
3
+ u
a
6
=
−
1
4
u
36
−
9
4
u
34
+ ··· −
1
2
u +
1
2
1
4
u
38
+
5
2
u
36
+ ··· + u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 2u
40
−4u
39
+ 22u
38
−40u
37
+ 114u
36
−196u
35
+ 364u
34
−604u
33
+ 786u
32
−1280u
31
+
1188u
30
− 1918u
29
+ 1256u
28
− 1996u
27
+ 904u
26
− 1290u
25
+ 434u
24
− 232u
23
+
184u
22
+ 468u
21
+ 136u
20
+ 520u
19
+ 84u
18
+ 194u
17
+ 4u
16
−120u
15
−4u
14
−258u
13
+
52u
12
− 234u
11
+ 84u
10
− 128u
9
+ 90u
8
− 30u
7
+ 64u
6
+ 2u
5
+ 24u
4
− 10u
3
− 4u
2
− 2u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
41
+ 2u
40
+ ··· + 5u − 1
c
2
u
41
+ 14u
40
+ ··· + u + 1
c
3
, c
8
u
41
+ 2u
40
+ ··· − 4u − 2
c
4
, c
7
u
41
− 2u
40
+ ··· − 24u − 16
c
6
, c
10
, c
11
u
41
− 2u
40
+ ··· − 7u − 1
c
9
u
41
− 22u
40
+ ··· + 8u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
41
− 14y
40
+ ··· + y −1
c
2
y
41
+ 34y
40
+ ··· + 737y −1
c
3
, c
8
y
41
+ 22y
40
+ ··· + 8y −4
c
4
, c
7
y
41
− 34y
40
+ ··· − 16448y −256
c
6
, c
10
, c
11
y
41
− 46y
40
+ ··· − 47y −1
c
9
y
41
− 6y
40
+ ··· + 160y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.503278 + 0.876227I
a = −1.01864 − 1.35377I
b = −0.184951 − 0.697202I
−1.85795 + 5.19311I −2.06190 − 8.35313I
u = −0.503278 − 0.876227I
a = −1.01864 + 1.35377I
b = −0.184951 + 0.697202I
−1.85795 − 5.19311I −2.06190 + 8.35313I
u = 0.595879 + 0.924278I
a = −1.261860 + 0.541147I
b = −0.683532 + 0.365288I
3.43135 − 8.38206I 3.46275 + 8.23571I
u = 0.595879 − 0.924278I
a = −1.261860 − 0.541147I
b = −0.683532 − 0.365288I
3.43135 + 8.38206I 3.46275 − 8.23571I
u = 0.012198 + 0.896132I
a = 1.24032 + 1.28683I
b = 0.765320 + 0.507138I
1.35320 − 1.46651I 7.22861 + 4.90542I
u = 0.012198 − 0.896132I
a = 1.24032 − 1.28683I
b = 0.765320 − 0.507138I
1.35320 + 1.46651I 7.22861 − 4.90542I
u = 0.662080 + 0.589897I
a = 0.314575 + 0.870590I
b = 0.143067 + 0.847335I
2.46953 + 3.52956I 2.12209 − 2.66433I
u = 0.662080 − 0.589897I
a = 0.314575 − 0.870590I
b = 0.143067 − 0.847335I
2.46953 − 3.52956I 2.12209 + 2.66433I
u = −0.860560 + 0.141859I
a = −0.542033 + 0.556076I
b = −0.51707 − 2.21472I
7.79923 − 9.18843I 3.80065 + 5.17633I
u = −0.860560 − 0.141859I
a = −0.542033 − 0.556076I
b = −0.51707 + 2.21472I
7.79923 + 9.18843I 3.80065 − 5.17633I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.861859 + 0.085768I
a = 0.673275 + 0.352479I
b = 0.26299 − 1.41656I
9.69240 + 2.90753I 6.01430 − 0.84016I
u = 0.861859 − 0.085768I
a = 0.673275 − 0.352479I
b = 0.26299 + 1.41656I
9.69240 − 2.90753I 6.01430 + 0.84016I
u = −0.053677 + 1.146820I
a = 0.37845 − 1.62203I
b = 0.571996 − 0.725064I
8.20281 + 2.94250I 9.69079 − 2.88880I
u = −0.053677 − 1.146820I
a = 0.37845 + 1.62203I
b = 0.571996 + 0.725064I
8.20281 − 2.94250I 9.69079 + 2.88880I
u = −0.547888 + 1.013950I
a = 1.410510 − 0.051509I
b = 1.097590 + 0.770395I
4.68586 + 3.37196I 6.53621 − 2.75945I
u = −0.547888 − 1.013950I
a = 1.410510 + 0.051509I
b = 1.097590 − 0.770395I
4.68586 − 3.37196I 6.53621 + 2.75945I
u = −0.423966 + 1.085320I
a = 0.889067 − 0.156517I
b = 1.100080 + 0.649290I
4.21251 + 3.60145I 9.92786 − 4.45844I
u = −0.423966 − 1.085320I
a = 0.889067 + 0.156517I
b = 1.100080 − 0.649290I
4.21251 − 3.60145I 9.92786 + 4.45844I
u = −0.683105 + 0.455196I
a = 0.331321 + 0.649818I
b = −0.675612 + 0.983763I
3.06237 + 1.36624I 3.26071 − 2.82351I
u = −0.683105 − 0.455196I
a = 0.331321 − 0.649818I
b = −0.675612 − 0.983763I
3.06237 − 1.36624I 3.26071 + 2.82351I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.800093 + 0.101020I
a = −0.166855 − 0.995960I
b = −0.60540 + 1.90379I
1.51266 + 5.04411I 1.12155 − 5.34007I
u = 0.800093 − 0.101020I
a = −0.166855 + 0.995960I
b = −0.60540 − 1.90379I
1.51266 − 5.04411I 1.12155 + 5.34007I
u = −0.500958 + 0.624025I
a = −0.051668 − 0.512607I
b = −0.589638 − 0.765864I
−2.57302 − 1.04199I −5.23599 + 1.20683I
u = −0.500958 − 0.624025I
a = −0.051668 + 0.512607I
b = −0.589638 + 0.765864I
−2.57302 + 1.04199I −5.23599 − 1.20683I
u = 0.465698 + 1.112630I
a = 1.14618 + 1.00180I
b = 1.113240 − 0.021535I
0.77731 − 3.72567I −1.87519 + 3.76789I
u = 0.465698 − 1.112630I
a = 1.14618 − 1.00180I
b = 1.113240 + 0.021535I
0.77731 + 3.72567I −1.87519 − 3.76789I
u = 0.405488 + 1.211450I
a = 1.31179 − 1.91130I
b = −0.56018 − 2.51021I
5.40261 + 0.90249I 5.70583 − 2.02309I
u = 0.405488 − 1.211450I
a = 1.31179 + 1.91130I
b = −0.56018 + 2.51021I
5.40261 − 0.90249I 5.70583 + 2.02309I
u = 0.497123 + 1.201300I
a = −0.84019 + 2.50910I
b = 1.40879 + 2.64017I
4.75069 − 9.79224I 4.22259 + 8.17334I
u = 0.497123 − 1.201300I
a = −0.84019 − 2.50910I
b = 1.40879 − 2.64017I
4.75069 + 9.79224I 4.22259 − 8.17334I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.373001 + 1.250030I
a = 1.33745 + 1.92758I
b = −0.56709 + 2.23140I
12.10740 − 4.99417I 8.18458 + 2.23571I
u = −0.373001 − 1.250030I
a = 1.33745 − 1.92758I
b = −0.56709 − 2.23140I
12.10740 + 4.99417I 8.18458 − 2.23571I
u = 0.410502 + 1.250480I
a = −0.850292 + 1.066310I
b = 0.602301 + 1.153840I
13.77940 − 1.51388I 9.75388 + 2.25000I
u = 0.410502 − 1.250480I
a = −0.850292 − 1.066310I
b = 0.602301 − 1.153840I
13.77940 + 1.51388I 9.75388 − 2.25000I
u = −0.526084 + 1.215280I
a = −1.27954 − 2.55127I
b = 0.88088 − 2.96625I
11.0125 + 14.2368I 6.63638 − 8.24449I
u = −0.526084 − 1.215280I
a = −1.27954 + 2.55127I
b = 0.88088 + 2.96625I
11.0125 − 14.2368I 6.63638 + 8.24449I
u = 0.502214 + 1.228010I
a = 0.78343 − 1.80145I
b = −0.41247 − 2.19576I
13.1184 − 7.8365I 8.99343 + 4.06275I
u = 0.502214 − 1.228010I
a = 0.78343 + 1.80145I
b = −0.41247 + 2.19576I
13.1184 + 7.8365I 8.99343 − 4.06275I
u = 0.526691 + 0.260209I
a = 0.059135 − 0.159552I
b = −0.942509 − 0.266856I
−1.66678 − 0.31810I −5.86833 + 0.81571I
u = 0.526691 − 0.260209I
a = 0.059135 + 0.159552I
b = −0.942509 + 0.266856I
−1.66678 + 0.31810I −5.86833 − 0.81571I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.534614
a = 1.27118
b = −0.415627
1.42690 6.75840
9
II. I
u
2
= h46u
4
a
2
+ 10u
4
a + · · · − 33a + 56, −u
4
a + 3u
4
+ · · · + 2a + 1, u
5
+
u
4
+ 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
1
=
a
−0.630137a
2
u
4
− 0.136986au
4
+ ··· + 0.452055a − 0.767123
a
3
=
1
−u
2
a
2
=
0.328767a
2
u
4
− 0.232877au
4
+ ··· + 0.0684932a + 1.09589
−0.739726a
2
u
4
+ 0.273973au
4
+ ··· + 1.09589a − 0.465753
a
8
=
u
u
a
10
=
u
3
u
3
+ u
a
5
=
−u
3
−u
4
− u
3
− u
2
− 1
a
7
=
−u
3
−u
4
− u
3
− u
2
− 1
a
11
=
0.328767a
2
u
4
− 0.232877au
4
+ ··· + 0.0684932a + 1.09589
−0.739726a
2
u
4
+ 0.273973au
4
+ ··· + 1.09589a − 0.465753
a
6
=
0.328767a
2
u
4
− 0.232877au
4
+ ··· + 0.0684932a + 1.09589
1.21918a
2
u
4
− 0.821918au
4
+ ··· − 1.28767a + 1.39726
a
6
=
0.328767a
2
u
4
− 0.232877au
4
+ ··· + 0.0684932a + 1.09589
1.21918a
2
u
4
− 0.821918au
4
+ ··· − 1.28767a + 1.39726
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
3
+ 4u
2
+ 4u + 6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
10
, c
11
u
15
− 5u
13
+ ··· − u + 1
c
2
u
15
+ 10u
14
+ ··· − u + 1
c
3
, c
8
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
3
c
4
, c
7
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
3
c
9
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
3
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
10
, c
11
y
15
− 10y
14
+ ··· − y − 1
c
2
y
15
− 10y
14
+ ··· − y − 1
c
3
, c
8
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
3
c
4
, c
7
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
3
c
9
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1)
3
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.339110 + 0.822375I
a = 0.987461 − 0.368120I
b = 0.550313 − 0.577492I
0.32910 − 1.53058I 2.51511 + 4.43065I
u = 0.339110 + 0.822375I
a = −0.519621 − 0.051702I
b = −1.65077 + 0.68097I
0.32910 − 1.53058I 2.51511 + 4.43065I
u = 0.339110 + 0.822375I
a = −0.21028 + 2.63515I
b = 0.480628 + 0.996348I
0.32910 − 1.53058I 2.51511 + 4.43065I
u = 0.339110 − 0.822375I
a = 0.987461 + 0.368120I
b = 0.550313 + 0.577492I
0.32910 + 1.53058I 2.51511 − 4.43065I
u = 0.339110 − 0.822375I
a = −0.519621 + 0.051702I
b = −1.65077 − 0.68097I
0.32910 + 1.53058I 2.51511 − 4.43065I
u = 0.339110 − 0.822375I
a = −0.21028 − 2.63515I
b = 0.480628 − 0.996348I
0.32910 + 1.53058I 2.51511 − 4.43065I
u = −0.766826
a = 0.584967 + 0.856589I
b = −0.40069 − 1.38845I
2.40108 3.48110
u = −0.766826
a = 0.584967 − 0.856589I
b = −0.40069 + 1.38845I
2.40108 3.48110
u = −0.766826
a = −0.429363
b = −1.63410
2.40108 3.48110
u = −0.455697 + 1.200150I
a = −0.33284 − 1.93140I
b = 1.58159 − 1.67595I
5.87256 + 4.40083I 6.74431 − 3.49859I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.455697 + 1.200150I
a = 1.19834 + 1.66866I
b = −0.22205 + 2.42247I
5.87256 + 4.40083I 6.74431 − 3.49859I
u = −0.455697 + 1.200150I
a = 1.50666 − 1.48655I
b = 1.47803 − 0.30819I
5.87256 + 4.40083I 6.74431 − 3.49859I
u = −0.455697 − 1.200150I
a = −0.33284 + 1.93140I
b = 1.58159 + 1.67595I
5.87256 − 4.40083I 6.74431 + 3.49859I
u = −0.455697 − 1.200150I
a = 1.19834 − 1.66866I
b = −0.22205 − 2.42247I
5.87256 − 4.40083I 6.74431 + 3.49859I
u = −0.455697 − 1.200150I
a = 1.50666 + 1.48655I
b = 1.47803 + 0.30819I
5.87256 − 4.40083I 6.74431 + 3.49859I
14
III. I
u
3
= hu
3
+ b + u − 1, u
3
− 2u
2
+ 2a − 4, u
4
+ 2u
2
+ 2i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
1
=
−
1
2
u
3
+ u
2
+ 2
−u
3
− u + 1
a
3
=
1
−u
2
a
2
=
−
1
2
u
3
+ u
2
+ 3
−u
3
− u
2
− u + 1
a
8
=
u
u
a
10
=
u
3
u
3
+ u
a
5
=
−1
u
2
a
7
=
−u
3
−u
3
− u
a
11
=
1
2
u
3
+ u
2
+ 2
1
a
6
=
−
1
2
u
3
+ u
2
+ 2
−u
3
− u + 1
a
6
=
−
1
2
u
3
+ u
2
+ 2
−u
3
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
+ 8
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
10
c
11
(u + 1)
4
c
3
, c
8
u
4
+ 2u
2
+ 2
c
4
, c
7
u
4
− 2u
2
+ 2
c
5
, c
6
(u − 1)
4
c
9
(u
2
− 2u + 2)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
10
, c
11
(y − 1)
4
c
3
, c
8
(y
2
+ 2y + 2)
2
c
4
, c
7
(y
2
− 2y + 2)
2
c
9
(y
2
+ 4)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.455090 + 1.098680I
a = 1.77689 + 1.32180I
b = 2.09868 − 0.45509I
2.46740 − 3.66386I 4.00000 + 4.00000I
u = 0.455090 − 1.098680I
a = 1.77689 − 1.32180I
b = 2.09868 + 0.45509I
2.46740 + 3.66386I 4.00000 − 4.00000I
u = −0.455090 + 1.098680I
a = 0.223113 − 0.678203I
b = −0.098684 − 0.455090I
2.46740 + 3.66386I 4.00000 − 4.00000I
u = −0.455090 − 1.098680I
a = 0.223113 + 0.678203I
b = −0.098684 + 0.455090I
2.46740 − 3.66386I 4.00000 + 4.00000I
18
IV. I
v
1
= ha, b + 1, v − 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
1
0
a
1
=
0
−1
a
3
=
1
0
a
2
=
1
−1
a
8
=
1
0
a
10
=
1
0
a
5
=
1
0
a
7
=
1
0
a
11
=
1
−1
a
6
=
0
1
a
6
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
, c
11
u − 1
c
2
, c
5
, c
6
u + 1
c
3
, c
4
, c
7
c
8
, c
9
u
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
10
, c
11
y − 1
c
3
, c
4
, c
7
c
8
, c
9
y
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −1.00000
0 0
22
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u + 1)
4
(u
15
− 5u
13
+ ··· − u + 1)(u
41
+ 2u
40
+ ··· + 5u − 1)
c
2
((u + 1)
5
)(u
15
+ 10u
14
+ ··· − u + 1)(u
41
+ 14u
40
+ ··· + u + 1)
c
3
, c
8
u(u
4
+ 2u
2
+ 2)(u
5
− u
4
+ ··· + u − 1)
3
(u
41
+ 2u
40
+ ··· − 4u − 2)
c
4
, c
7
u(u
4
− 2u
2
+ 2)(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
3
· (u
41
− 2u
40
+ ··· − 24u − 16)
c
5
((u − 1)
4
)(u + 1)(u
15
− 5u
13
+ ··· − u + 1)(u
41
+ 2u
40
+ ··· + 5u − 1)
c
6
((u − 1)
4
)(u + 1)(u
15
− 5u
13
+ ··· − u + 1)(u
41
− 2u
40
+ ··· − 7u − 1)
c
9
u(u
2
− 2u + 2)
2
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
3
· (u
41
− 22u
40
+ ··· + 8u + 4)
c
10
, c
11
(u − 1)(u + 1)
4
(u
15
− 5u
13
+ ··· − u + 1)(u
41
− 2u
40
+ ··· − 7u − 1)
23
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
((y − 1)
5
)(y
15
− 10y
14
+ ··· − y − 1)(y
41
− 14y
40
+ ··· + y − 1)
c
2
((y − 1)
5
)(y
15
− 10y
14
+ ··· − y − 1)(y
41
+ 34y
40
+ ··· + 737y − 1)
c
3
, c
8
y(y
2
+ 2y + 2)
2
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
3
· (y
41
+ 22y
40
+ ··· + 8y − 4)
c
4
, c
7
y(y
2
− 2y + 2)
2
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
3
· (y
41
− 34y
40
+ ··· − 16448y − 256)
c
6
, c
10
, c
11
((y − 1)
5
)(y
15
− 10y
14
+ ··· − y − 1)(y
41
− 46y
40
+ ··· − 47y − 1)
c
9
y(y
2
+ 4)
2
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1)
3
· (y
41
− 6y
40
+ ··· + 160y − 16)
24
