11a
31
(K11a
31
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 10 11 3 5 6 9 8
Solving Sequence
5,10 2,6
4 1 9 11 7 3 8
c
5
c
4
c
1
c
9
c
10
c
6
c
3
c
8
c
2
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
65
+ u
64
+ ··· + b − u, −u
65
+ u
64
+ ··· + a − 1, u
67
− 2u
66
+ ··· − 4u
2
+ 1i
I
u
2
= hb + 1, −u
3
+ u
2
+ a − u + 2, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 72 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
65
+u
64
+· · ·+b−u, −u
65
+u
64
+· · ·+a−1, u
67
−2u
66
+· · ·−4u
2
+1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
2
=
u
65
− u
64
+ ··· − 2u + 1
u
65
− u
64
+ ··· − 2u
2
+ u
a
6
=
1
−u
2
a
4
=
2u
65
− 2u
64
+ ··· − 2u + 2
u
65
− u
64
+ ··· + 5u
3
− 3u
2
a
1
=
−u
11
− 2u
9
− 2u
7
− u
3
−u
11
− 3u
9
− 4u
7
− u
5
+ u
3
+ u
a
9
=
−u
u
3
+ u
a
11
=
−u
3
u
5
+ u
3
+ u
a
7
=
−u
6
− u
4
+ 1
u
8
+ 2u
6
+ 2u
4
a
3
=
−u
63
+ u
62
+ ··· − u + 1
u
65
− u
64
+ ··· − u
2
+ u
a
8
=
u
3
u
3
+ u
a
8
=
u
3
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
66
+ 13u
65
+ ··· + u − 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
67
− 6u
66
+ ··· − 6u + 1
c
2
u
67
+ 32u
66
+ ··· − 6u + 1
c
3
, c
7
u
67
+ u
66
+ ··· + 96u + 32
c
5
, c
9
u
67
− 2u
66
+ ··· − 4u
2
+ 1
c
6
, c
8
u
67
+ 2u
66
+ ··· + 78u + 9
c
10
u
67
+ 36u
66
+ ··· + 8u − 1
c
11
u
67
− 8u
66
+ ··· + 2798u + 53
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
67
− 32y
66
+ ··· − 6y −1
c
2
y
67
+ 12y
66
+ ··· − 266y −1
c
3
, c
7
y
67
+ 33y
66
+ ··· − 14848y −1024
c
5
, c
9
y
67
+ 36y
66
+ ··· + 8y −1
c
6
, c
8
y
67
− 52y
66
+ ··· − 360y −81
c
10
y
67
− 8y
66
+ ··· + 124y −1
c
11
y
67
+ 8y
66
+ ··· + 6484936y −2809
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.564990 + 0.825170I
a = −1.000610 + 0.092379I
b = 0.481444 − 0.849559I
4.87488 − 4.14687I −1.59935 + 4.64549I
u = −0.564990 − 0.825170I
a = −1.000610 − 0.092379I
b = 0.481444 + 0.849559I
4.87488 + 4.14687I −1.59935 − 4.64549I
u = 0.511350 + 0.819428I
a = −0.40478 − 2.45910I
b = −0.887590 + 0.499549I
0.03256 + 4.08481I −5.93835 − 7.04941I
u = 0.511350 − 0.819428I
a = −0.40478 + 2.45910I
b = −0.887590 − 0.499549I
0.03256 − 4.08481I −5.93835 + 7.04941I
u = −0.569629 + 0.864915I
a = 1.27940 − 2.06653I
b = 1.098240 + 0.652944I
3.02110 − 9.72497I −4.76527 + 9.27372I
u = −0.569629 − 0.864915I
a = 1.27940 + 2.06653I
b = 1.098240 − 0.652944I
3.02110 + 9.72497I −4.76527 − 9.27372I
u = 0.246240 + 1.034280I
a = 0.823566 + 0.817173I
b = 0.575942 + 0.558676I
−0.281369 + 0.970663I 0
u = 0.246240 − 1.034280I
a = 0.823566 − 0.817173I
b = 0.575942 − 0.558676I
−0.281369 − 0.970663I 0
u = 0.119094 + 1.072640I
a = 2.44789 − 0.02818I
b = 1.039730 − 0.556055I
−1.77198 + 5.50921I 0
u = 0.119094 − 1.072640I
a = 2.44789 + 0.02818I
b = 1.039730 + 0.556055I
−1.77198 − 5.50921I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.052245 + 0.915458I
a = −2.62750 + 1.03290I
b = −1.058310 − 0.288428I
−3.49198 − 1.02780I −15.4724 + 0.3169I
u = −0.052245 − 0.915458I
a = −2.62750 − 1.03290I
b = −1.058310 + 0.288428I
−3.49198 + 1.02780I −15.4724 − 0.3169I
u = −0.465256 + 0.785906I
a = −0.92627 + 1.18008I
b = −1.225020 + 0.046584I
−1.16301 − 1.95072I −3.00568 + 5.39206I
u = −0.465256 − 0.785906I
a = −0.92627 − 1.18008I
b = −1.225020 − 0.046584I
−1.16301 + 1.95072I −3.00568 − 5.39206I
u = −0.574831 + 0.702829I
a = 0.276564 − 1.126500I
b = 0.534420 + 0.823507I
5.22376 − 0.38517I −0.51661 + 2.40952I
u = −0.574831 − 0.702829I
a = 0.276564 + 1.126500I
b = 0.534420 − 0.823507I
5.22376 + 0.38517I −0.51661 − 2.40952I
u = −0.592161 + 0.649495I
a = −0.159409 + 0.601501I
b = 1.063280 − 0.655308I
3.63203 + 5.13427I −3.00083 − 2.99523I
u = −0.592161 − 0.649495I
a = −0.159409 − 0.601501I
b = 1.063280 + 0.655308I
3.63203 − 5.13427I −3.00083 + 2.99523I
u = 0.493599 + 0.712589I
a = 0.989105 + 0.915694I
b = −0.797853 − 0.467895I
0.345703 + 0.068999I −4.55173 − 0.43344I
u = 0.493599 − 0.712589I
a = 0.989105 − 0.915694I
b = −0.797853 + 0.467895I
0.345703 − 0.068999I −4.55173 + 0.43344I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.249006 + 0.819006I
a = 0.686833 + 0.149955I
b = 0.113095 + 0.211783I
−0.492874 + 1.272410I −5.34668 − 4.93990I
u = 0.249006 − 0.819006I
a = 0.686833 − 0.149955I
b = 0.113095 − 0.211783I
−0.492874 − 1.272410I −5.34668 + 4.93990I
u = 0.816854 + 0.158141I
a = 1.31512 − 1.20053I
b = 1.150520 + 0.623562I
−0.49031 − 10.48350I −6.68771 + 6.96472I
u = 0.816854 − 0.158141I
a = 1.31512 + 1.20053I
b = 1.150520 − 0.623562I
−0.49031 + 10.48350I −6.68771 − 6.96472I
u = −0.819060 + 0.039508I
a = 1.247840 + 0.239436I
b = 0.943939 + 0.369724I
−3.90258 − 1.39316I −7.44622 + 4.95368I
u = −0.819060 − 0.039508I
a = 1.247840 − 0.239436I
b = 0.943939 − 0.369724I
−3.90258 + 1.39316I −7.44622 − 4.95368I
u = 0.491373 + 1.074670I
a = 0.250948 + 0.870006I
b = 0.912328 + 0.663370I
0.415223 + 0.749566I 0
u = 0.491373 − 1.074670I
a = 0.250948 − 0.870006I
b = 0.912328 − 0.663370I
0.415223 − 0.749566I 0
u = 0.787826 + 0.169798I
a = −0.403477 + 0.028686I
b = 0.374311 − 0.872073I
1.83934 − 4.96300I −3.50692 + 3.21590I
u = 0.787826 − 0.169798I
a = −0.403477 − 0.028686I
b = 0.374311 + 0.872073I
1.83934 + 4.96300I −3.50692 − 3.21590I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.771054 + 0.136932I
a = −0.93208 − 1.67887I
b = −1.010020 + 0.515425I
−2.81565 + 4.36823I −8.36975 − 4.25487I
u = −0.771054 − 0.136932I
a = −0.93208 + 1.67887I
b = −1.010020 − 0.515425I
−2.81565 − 4.36823I −8.36975 + 4.25487I
u = 0.506993 + 1.122470I
a = 1.55101 + 0.80025I
b = 0.704157 − 0.772274I
1.05507 + 6.16799I 0
u = 0.506993 − 1.122470I
a = 1.55101 − 0.80025I
b = 0.704157 + 0.772274I
1.05507 − 6.16799I 0
u = 0.751013 + 0.111032I
a = −1.48016 + 0.29513I
b = −1.254480 + 0.179113I
−3.63648 − 1.86851I −7.74469 + 3.58479I
u = 0.751013 − 0.111032I
a = −1.48016 − 0.29513I
b = −1.254480 − 0.179113I
−3.63648 + 1.86851I −7.74469 − 3.58479I
u = −0.423713 + 1.167870I
a = −0.210888 + 0.121533I
b = −0.351913 − 0.524199I
−4.77435 − 3.67797I 0
u = −0.423713 − 1.167870I
a = −0.210888 − 0.121533I
b = −0.351913 + 0.524199I
−4.77435 + 3.67797I 0
u = 0.360436 + 1.193060I
a = 0.685565 − 0.715503I
b = 0.331641 − 0.843858I
−2.23668 − 1.18578I 0
u = 0.360436 − 1.193060I
a = 0.685565 + 0.715503I
b = 0.331641 + 0.843858I
−2.23668 + 1.18578I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.385671 + 1.191060I
a = −1.81898 − 0.21940I
b = −1.047790 + 0.496161I
−6.70689 + 0.47788I 0
u = −0.385671 − 1.191060I
a = −1.81898 + 0.21940I
b = −1.047790 − 0.496161I
−6.70689 − 0.47788I 0
u = 0.402865 + 1.186580I
a = −2.97982 − 0.67442I
b = −1.251850 + 0.222499I
−7.37230 + 2.08523I 0
u = 0.402865 − 1.186580I
a = −2.97982 + 0.67442I
b = −1.251850 − 0.222499I
−7.37230 − 2.08523I 0
u = −0.481238 + 1.165490I
a = 0.648241 + 0.407995I
b = −0.501049 + 0.552756I
−4.35980 − 4.63647I 0
u = −0.481238 − 1.165490I
a = 0.648241 − 0.407995I
b = −0.501049 − 0.552756I
−4.35980 + 4.63647I 0
u = 0.363456 + 1.217370I
a = 2.20761 + 0.35528I
b = 1.152000 + 0.602107I
−4.67059 − 6.53918I 0
u = 0.363456 − 1.217370I
a = 2.20761 − 0.35528I
b = 1.152000 − 0.602107I
−4.67059 + 6.53918I 0
u = 0.678065 + 0.264736I
a = 0.241702 − 0.802492I
b = 0.639652 + 0.752359I
3.53541 − 1.62116I −1.50598 + 2.39002I
u = 0.678065 − 0.264736I
a = 0.241702 + 0.802492I
b = 0.639652 − 0.752359I
3.53541 + 1.62116I −1.50598 − 2.39002I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.636846 + 0.349077I
a = 0.126882 + 0.689081I
b = 0.982683 − 0.638281I
2.50124 + 3.65316I −3.04825 − 3.77451I
u = 0.636846 − 0.349077I
a = 0.126882 − 0.689081I
b = 0.982683 + 0.638281I
2.50124 − 3.65316I −3.04825 + 3.77451I
u = 0.493300 + 1.180590I
a = −2.28193 − 1.49318I
b = −1.286340 − 0.182808I
−6.72971 + 6.47733I 0
u = 0.493300 − 1.180590I
a = −2.28193 + 1.49318I
b = −1.286340 + 0.182808I
−6.72971 − 6.47733I 0
u = −0.504577 + 1.182750I
a = −2.16655 + 1.99170I
b = −1.022760 − 0.541798I
−5.86773 − 9.08868I 0
u = −0.504577 − 1.182750I
a = −2.16655 − 1.99170I
b = −1.022760 + 0.541798I
−5.86773 + 9.08868I 0
u = 0.518615 + 1.181260I
a = −0.854148 + 0.948142I
b = 0.360963 + 0.899604I
−1.13304 + 9.79846I 0
u = 0.518615 − 1.181260I
a = −0.854148 − 0.948142I
b = 0.360963 − 0.899604I
−1.13304 − 9.79846I 0
u = −0.434695 + 1.222970I
a = 2.46804 − 0.22231I
b = 0.982398 + 0.369460I
−7.66696 − 5.81189I 0
u = −0.434695 − 1.222970I
a = 2.46804 + 0.22231I
b = 0.982398 − 0.369460I
−7.66696 + 5.81189I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.521787 + 1.193990I
a = 2.69334 + 1.63831I
b = 1.164330 − 0.627272I
−3.5558 + 15.4016I 0
u = 0.521787 − 1.193990I
a = 2.69334 − 1.63831I
b = 1.164330 + 0.627272I
−3.5558 − 15.4016I 0
u = −0.473515 + 1.216160I
a = 1.73934 − 1.24342I
b = 0.939042 − 0.333718I
−7.39058 − 3.26134I 0
u = −0.473515 − 1.216160I
a = 1.73934 + 1.24342I
b = 0.939042 + 0.333718I
−7.39058 + 3.26134I 0
u = −0.680235 + 0.090499I
a = 0.733959 + 0.447029I
b = −0.481544 − 0.437537I
−1.341860 + 0.241306I −6.37372 + 0.86588I
u = −0.680235 − 0.090499I
a = 0.733959 − 0.447029I
b = −0.481544 + 0.437537I
−1.341860 − 0.241306I −6.37372 − 0.86588I
u = −0.311701
a = 1.66731
b = −0.735196
−1.10322 −8.76950
11
II. I
u
2
= hb + 1, −u
3
+ u
2
+ a − u + 2, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
2
=
u
3
− u
2
+ u − 2
−1
a
6
=
1
−u
2
a
4
=
u
3
− u
2
+ u − 1
−1
a
1
=
−1
0
a
9
=
−u
u
3
+ u
a
11
=
−u
3
u
4
− u
3
+ u
2
+ 1
a
7
=
u
3
u
3
+ u
a
3
=
u
3
− u
2
+ u − 1
−1
a
8
=
u
3
u
3
+ u
a
8
=
u
3
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
4
+ u
3
+ 2u − 12
12
(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
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
6
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
8
, c
11
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
9
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
10
u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1
13
(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
9
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
6
, c
8
, c
11
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
10
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.339110 + 0.822375I
a = −1.12878 + 1.10766I
b = −1.00000
−1.97403 − 1.53058I −12.02124 + 2.62456I
u = −0.339110 − 0.822375I
a = −1.12878 − 1.10766I
b = −1.00000
−1.97403 + 1.53058I −12.02124 − 2.62456I
u = 0.766826
a = −1.37029
b = −1.00000
−4.04602 −9.32390
u = 0.455697 + 1.200150I
a = −2.18608 − 0.87465I
b = −1.00000
−7.51750 + 4.40083I −12.31681 − 3.97407I
u = 0.455697 − 1.200150I
a = −2.18608 + 0.87465I
b = −1.00000
−7.51750 − 4.40083I −12.31681 + 3.97407I
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
67
− 6u
66
+ ··· − 6u + 1)
c
2
((u + 1)
5
)(u
67
+ 32u
66
+ ··· − 6u + 1)
c
3
, c
7
u
5
(u
67
+ u
66
+ ··· + 96u + 32)
c
4
((u + 1)
5
)(u
67
− 6u
66
+ ··· − 6u + 1)
c
5
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)(u
67
− 2u
66
+ ··· − 4u
2
+ 1)
c
6
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
67
+ 2u
66
+ ··· + 78u + 9)
c
8
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
67
+ 2u
66
+ ··· + 78u + 9)
c
9
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
67
− 2u
66
+ ··· − 4u
2
+ 1)
c
10
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)(u
67
+ 36u
66
+ ··· + 8u − 1)
c
11
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
67
− 8u
66
+ ··· + 2798u + 53)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
5
)(y
67
− 32y
66
+ ··· − 6y −1)
c
2
((y −1)
5
)(y
67
+ 12y
66
+ ··· − 266y −1)
c
3
, c
7
y
5
(y
67
+ 33y
66
+ ··· − 14848y −1024)
c
5
, c
9
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
67
+ 36y
66
+ ··· + 8y −1)
c
6
, c
8
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
67
− 52y
66
+ ··· − 360y −81)
c
10
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)(y
67
− 8y
66
+ ··· + 124y −1)
c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
67
+ 8y
66
+ ··· + 6484936y −2809)
17
