11n
4
(K11n
4
)
A knot diagram
1
Linearized knot diagam
5 1 7 2 3 9 4 11 6 1 9
Solving Sequence
1,5
2 3 6
4,9
11 8 7 10
c
1
c
2
c
5
c
4
c
11
c
8
c
7
c
10
c
3
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h9579649u
28
+ 28051230u
27
+ ··· + 44628149b − 26156718,
− 402165u
28
− 18012803u
27
+ ··· + 44628149a − 25970340, u
29
+ 2u
28
+ ··· − u + 1i
I
u
2
= hb − 1, u
4
− u
3
+ 2u
2
+ a − u + 1, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 34 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
= h9.58 × 10
6
u
28
+ 2.81 × 10
7
u
27
+ · · · + 4.46 × 10
7
b − 2.62 × 10
7
, −4.02 ×
10
5
u
28
− 1.80 × 10
7
u
27
+ · · · + 4.46 × 10
7
a − 2.60 × 10
7
, u
29
+ 2u
28
+ · · · − u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
3
=
u
2
+ 1
−u
2
a
6
=
−u
5
− 2u
3
− u
u
5
+ u
3
+ u
a
4
=
−u
u
3
+ u
a
9
=
0.00901146u
28
+ 0.403620u
27
+ ··· + 0.0183714u + 0.581927
−0.214655u
28
− 0.628555u
27
+ ··· + 0.800456u + 0.586104
a
11
=
−0.250701u
28
− 0.243034u
27
+ ··· + 0.726970u + 1.25839
−0.141381u
28
− 0.485781u
27
+ ··· + 0.798176u + 0.655586
a
8
=
1.26592u
28
+ 2.07814u
27
+ ··· − 1.78235u − 0.0609626
−0.853451u
28
− 1.71445u
27
+ ··· + 0.995441u − 0.861036
a
7
=
0.667701u
28
+ 1.07191u
27
+ ··· − 2.17896u − 0.0640826
−0.263489u
28
− 0.531459u
27
+ ··· + 0.603619u − 0.667701
a
10
=
−0.109320u
28
+ 0.242748u
27
+ ··· − 0.0712062u + 0.602809
−0.141381u
28
− 0.485781u
27
+ ··· + 0.798176u + 0.655586
a
10
=
−0.109320u
28
+ 0.242748u
27
+ ··· − 0.0712062u + 0.602809
−0.141381u
28
− 0.485781u
27
+ ··· + 0.798176u + 0.655586
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
2569949
44628149
u
28
−
167545923
44628149
u
27
+ ··· −
46388860
44628149
u +
58164951
44628149
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
29
+ 2u
28
+ ··· − u + 1
c
2
u
29
+ 12u
28
+ ··· − 5u − 1
c
3
, c
7
u
29
+ 2u
28
+ ··· + u + 1
c
5
u
29
− 2u
28
+ ··· − 65u + 17
c
6
, c
9
u
29
− 5u
28
+ ··· + 24u
2
+ 32
c
8
, c
11
u
29
+ 6u
28
+ ··· + 5u + 1
c
10
u
29
− 36u
28
+ ··· − 183u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
29
+ 12y
28
+ ··· − 5y −1
c
2
y
29
+ 12y
28
+ ··· − 89y −1
c
3
, c
7
y
29
+ 30y
27
+ ··· − 5y −1
c
5
y
29
+ 12y
28
+ ··· − 13285y −289
c
6
, c
9
y
29
+ 33y
28
+ ··· − 1536y −1024
c
8
, c
11
y
29
− 36y
28
+ ··· − 183y −1
c
10
y
29
− 80y
28
+ ··· + 16377y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.387233 + 0.859940I
a = 0.839842 − 0.200433I
b = −0.0183680 − 0.0952600I
−0.34137 + 1.65783I −2.51721 − 4.37356I
u = 0.387233 − 0.859940I
a = 0.839842 + 0.200433I
b = −0.0183680 + 0.0952600I
−0.34137 − 1.65783I −2.51721 + 4.37356I
u = 0.525029 + 0.781903I
a = −1.45815 − 1.69363I
b = 0.960834 − 0.144408I
1.78487 + 1.57609I 0.9666 − 16.5900I
u = 0.525029 − 0.781903I
a = −1.45815 + 1.69363I
b = 0.960834 + 0.144408I
1.78487 − 1.57609I 0.9666 + 16.5900I
u = −0.654583 + 0.675856I
a = −0.197062 − 0.780511I
b = 0.929333 + 1.022590I
3.12622 + 1.43345I 4.04144 − 2.82912I
u = −0.654583 − 0.675856I
a = −0.197062 + 0.780511I
b = 0.929333 − 1.022590I
3.12622 − 1.43345I 4.04144 + 2.82912I
u = −0.925881 + 0.518414I
a = 1.81441 + 0.23795I
b = −1.71997 − 0.32324I
11.74770 + 6.59261I 3.06245 − 2.55361I
u = −0.925881 − 0.518414I
a = 1.81441 − 0.23795I
b = −1.71997 + 0.32324I
11.74770 − 6.59261I 3.06245 + 2.55361I
u = 0.937398 + 0.500154I
a = 1.79802 − 0.06455I
b = −1.70627 − 0.02414I
11.61340 + 1.70244I 3.44440 − 1.84569I
u = 0.937398 − 0.500154I
a = 1.79802 + 0.06455I
b = −1.70627 + 0.02414I
11.61340 − 1.70244I 3.44440 + 1.84569I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.662767 + 0.848656I
a = −1.81457 − 1.32278I
b = 1.81677 − 0.13637I
5.01547 − 2.56835I 6.29777 + 3.45072I
u = −0.662767 − 0.848656I
a = −1.81457 + 1.32278I
b = 1.81677 + 0.13637I
5.01547 + 2.56835I 6.29777 − 3.45072I
u = 0.567900 + 0.933831I
a = −0.28660 + 1.95388I
b = 0.704163 + 0.280595I
1.23273 + 2.84215I 0.371066 + 0.581587I
u = 0.567900 − 0.933831I
a = −0.28660 − 1.95388I
b = 0.704163 − 0.280595I
1.23273 − 2.84215I 0.371066 − 0.581587I
u = −0.043975 + 0.873551I
a = 1.118610 + 0.586417I
b = 0.157858 + 0.616140I
−1.21438 + 1.50101I −6.11641 − 3.93982I
u = −0.043975 − 0.873551I
a = 1.118610 − 0.586417I
b = 0.157858 − 0.616140I
−1.21438 − 1.50101I −6.11641 + 3.93982I
u = −0.637441 + 0.973302I
a = −1.34495 − 0.55786I
b = 0.69848 − 1.23040I
2.23506 − 6.49074I 1.39267 + 8.34462I
u = −0.637441 − 0.973302I
a = −1.34495 + 0.55786I
b = 0.69848 + 1.23040I
2.23506 + 6.49074I 1.39267 − 8.34462I
u = −0.461488 + 1.163620I
a = 0.358541 + 0.419840I
b = −0.655831 − 0.154039I
−4.83578 − 4.15032I −10.94337 + 1.86325I
u = −0.461488 − 1.163620I
a = 0.358541 − 0.419840I
b = −0.655831 + 0.154039I
−4.83578 + 4.15032I −10.94337 − 1.86325I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.017652 + 1.279430I
a = −0.248173 − 0.176688I
b = −1.56169 − 0.18501I
4.96946 + 4.25609I −1.11871 − 2.71437I
u = 0.017652 − 1.279430I
a = −0.248173 + 0.176688I
b = −1.56169 + 0.18501I
4.96946 − 4.25609I −1.11871 + 2.71437I
u = −0.697556 + 1.121820I
a = 1.35862 + 1.61660I
b = −1.69014 + 0.41795I
9.9003 − 12.5531I 0.87648 + 6.84593I
u = −0.697556 − 1.121820I
a = 1.35862 − 1.61660I
b = −1.69014 − 0.41795I
9.9003 + 12.5531I 0.87648 − 6.84593I
u = 0.697068 + 1.137050I
a = 0.98961 − 1.53355I
b = −1.66331 − 0.08384I
9.66493 + 4.29038I 1.47355 − 2.52385I
u = 0.697068 − 1.137050I
a = 0.98961 + 1.53355I
b = −1.66331 + 0.08384I
9.66493 − 4.29038I 1.47355 + 2.52385I
u = −0.659229
a = 1.11847
b = −0.442580
−1.67720 −6.86830
u = 0.281024 + 0.265729I
a = 1.012610 − 0.151234I
b = 0.969423 + 0.291280I
1.86776 + 0.92254I 4.20343 − 0.65997I
u = 0.281024 − 0.265729I
a = 1.012610 + 0.151234I
b = 0.969423 − 0.291280I
1.86776 − 0.92254I 4.20343 + 0.65997I
7
II. I
u
2
= hb − 1, u
4
− u
3
+ 2u
2
+ a − u + 1, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
3
=
u
2
+ 1
−u
2
a
6
=
−u
4
− u
2
− 1
u
4
− u
3
+ u
2
+ 1
a
4
=
−u
u
3
+ u
a
9
=
−u
4
+ u
3
− 2u
2
+ u − 1
1
a
11
=
−u
4
+ u
3
− 2u
2
+ u
1
a
8
=
−1
0
a
7
=
−u
4
− u
2
− 1
u
4
− u
3
+ u
2
+ 1
a
10
=
−u
4
+ u
3
− 2u
2
+ u − 1
1
a
10
=
−u
4
+ u
3
− 2u
2
+ u − 1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
4
+ 3u
3
− 4u
2
+ 8u − 3
8
(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
+ 3u
4
+ 4u
3
+ u
2
− u − 1
c
3
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
4
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
5
, c
7
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
6
, c
9
u
5
c
8
, c
10
(u + 1)
5
c
11
(u − 1)
5
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
2
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
c
3
, c
5
, c
7
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
6
, c
9
y
5
c
8
, c
10
, c
11
(y −1)
5
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.339110 + 0.822375I
a = 0.428550 + 1.039280I
b = 1.00000
1.31583 − 1.53058I −1.50865 + 9.87103I
u = −0.339110 − 0.822375I
a = 0.428550 − 1.039280I
b = 1.00000
1.31583 + 1.53058I −1.50865 − 9.87103I
u = 0.766826
a = −1.30408
b = 1.00000
−0.756147 3.17260
u = 0.455697 + 1.200150I
a = −0.276511 + 0.728237I
b = 1.00000
−4.22763 + 4.40083I 0.92237 − 5.80708I
u = 0.455697 − 1.200150I
a = −0.276511 − 0.728237I
b = 1.00000
−4.22763 − 4.40083I 0.92237 + 5.80708I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)(u
29
+ 2u
28
+ ··· − u + 1)
c
2
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)(u
29
+ 12u
28
+ ··· − 5u − 1)
c
3
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
29
+ 2u
28
+ ··· + u + 1)
c
4
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
29
+ 2u
28
+ ··· − u + 1)
c
5
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
29
− 2u
28
+ ··· − 65u + 17)
c
6
, c
9
u
5
(u
29
− 5u
28
+ ··· + 24u
2
+ 32)
c
7
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
29
+ 2u
28
+ ··· + u + 1)
c
8
((u + 1)
5
)(u
29
+ 6u
28
+ ··· + 5u + 1)
c
10
((u + 1)
5
)(u
29
− 36u
28
+ ··· − 183u − 1)
c
11
((u − 1)
5
)(u
29
+ 6u
28
+ ··· + 5u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
29
+ 12y
28
+ ··· − 5y −1)
c
2
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)(y
29
+ 12y
28
+ ··· − 89y −1)
c
3
, c
7
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
29
+ 30y
27
+ ··· − 5y −1)
c
5
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
29
+ 12y
28
+ ··· − 13285y −289)
c
6
, c
9
y
5
(y
29
+ 33y
28
+ ··· − 1536y −1024)
c
8
, c
11
((y −1)
5
)(y
29
− 36y
28
+ ··· − 183y −1)
c
10
((y −1)
5
)(y
29
− 80y
28
+ ··· + 16377y −1)
13
