11n
90
(K11n
90
)
A knot diagram
1
Linearized knot diagam
6 1 9 8 2 3 11 4 5 1 8
Solving Sequence
1,6
2 3
5,8
4 11 7 10 9
c
1
c
2
c
5
c
4
c
11
c
7
c
10
c
9
c
3
, c
6
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−1238113u
25
+ 2413455u
24
+ ··· + 2221939b + 1586090,
4519597u
25
− 816584u
24
+ ··· + 13331634a − 13110217, u
26
− 2u
25
+ ··· − u + 3i
I
u
2
= hb − 1, a
2
− 2au + 2a + u − 2, u
2
− u + 1i
I
u
3
= hb + 1, a − u − 1, u
2
+ u + 1i
* 3 irreducible components of dim
C
= 0, with total 32 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−1.24 × 10
6
u
25
+ 2.41 × 10
6
u
24
+ · · · + 2.22 × 10
6
b + 1.59 × 10
6
, 4.52 ×
10
6
u
25
− 8.17 × 10
5
u
24
+ · · · + 1.33 × 10
7
a − 1.31 × 10
7
, u
26
− 2u
25
+ · · · − u + 3i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
−u
2
a
3
=
u
2
+ 1
−u
2
a
5
=
−u
u
3
+ u
a
8
=
−0.339013u
25
+ 0.0612516u
24
+ ··· + 0.0332859u + 0.983392
0.557222u
25
− 1.08619u
24
+ ··· + 1.41616u − 0.713831
a
4
=
−0.254668u
25
+ 0.460389u
24
+ ··· − 3.72725u − 2.65550
0.0489476u
25
− 0.702694u
24
+ ··· + 2.91017u − 0.764005
a
11
=
0.321877u
25
+ 0.0594569u
24
+ ··· − 3.45009u + 0.683981
−0.681651u
25
+ 1.22699u
24
+ ··· − 0.0147380u + 0.770149
a
7
=
−u
5
− 2u
3
− u
u
5
+ u
3
+ u
a
10
=
−0.359774u
25
+ 1.28645u
24
+ ··· − 3.46483u + 1.45413
−0.681651u
25
+ 1.22699u
24
+ ··· − 0.0147380u + 0.770149
a
9
=
0.237944u
25
+ 0.0813342u
24
+ ··· − 2.32111u + 1.17821
−0.616774u
25
+ 1.26410u
24
+ ··· + 0.644379u + 1.01704
a
9
=
0.237944u
25
+ 0.0813342u
24
+ ··· − 2.32111u + 1.17821
−0.616774u
25
+ 1.26410u
24
+ ··· + 0.644379u + 1.01704
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
1151955
2221939
u
25
+
814709
2221939
u
24
+ ··· −
5153343
2221939
u −
28973115
2221939
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
26
− 2u
25
+ ··· − u + 3
c
2
u
26
+ 16u
25
+ ··· − 43u + 9
c
3
, c
4
, c
8
u
26
+ u
25
+ ··· − 8u − 4
c
6
u
26
+ 2u
25
+ ··· − 13u + 3
c
7
, c
11
u
26
+ 3u
25
+ ··· + 22u − 3
c
9
u
26
− u
25
+ ··· − 32u − 4
c
10
u
26
+ 33u
25
+ ··· + 64u + 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
26
+ 16y
25
+ ··· − 43y + 9
c
2
y
26
− 8y
25
+ ··· − 7123y + 81
c
3
, c
4
, c
8
y
26
+ 21y
25
+ ··· + 64y + 16
c
6
y
26
− 32y
25
+ ··· − 187y + 9
c
7
, c
11
y
26
− 33y
25
+ ··· − 64y + 9
c
9
y
26
− 39y
25
+ ··· − 128y + 16
c
10
y
26
− 73y
25
+ ··· + 35108y + 81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.987320 + 0.168214I
a = 1.297460 + 0.513052I
b = −1.63497 − 0.20181I
−5.22414 − 5.39338I −6.45106 + 2.82273I
u = 0.987320 − 0.168214I
a = 1.297460 − 0.513052I
b = −1.63497 + 0.20181I
−5.22414 + 5.39338I −6.45106 − 2.82273I
u = −1.01037
a = −1.32902
b = 1.68442
−9.37437 −9.45940
u = −0.541900 + 0.798242I
a = −1.229580 − 0.630085I
b = 0.190153 + 0.181187I
4.95516 − 2.19764I 0.54342 + 3.86213I
u = −0.541900 − 0.798242I
a = −1.229580 + 0.630085I
b = 0.190153 − 0.181187I
4.95516 + 2.19764I 0.54342 − 3.86213I
u = 0.280901 + 0.919746I
a = 0.066362 − 0.266060I
b = 0.270359 + 0.442643I
−0.60039 + 1.42912I −6.05587 − 3.68708I
u = 0.280901 − 0.919746I
a = 0.066362 + 0.266060I
b = 0.270359 − 0.442643I
−0.60039 − 1.42912I −6.05587 + 3.68708I
u = −0.086149 + 0.939073I
a = 1.12447 + 1.41361I
b = 1.170170 − 0.263604I
1.87196 − 0.46648I −9.69334 − 0.39377I
u = −0.086149 − 0.939073I
a = 1.12447 − 1.41361I
b = 1.170170 + 0.263604I
1.87196 + 0.46648I −9.69334 + 0.39377I
u = −0.714585 + 0.848170I
a = 1.111130 + 0.871548I
b = −1.336900 + 0.007719I
−0.16795 − 2.70526I −6.45261 + 3.54399I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.714585 − 0.848170I
a = 1.111130 − 0.871548I
b = −1.336900 − 0.007719I
−0.16795 + 2.70526I −6.45261 − 3.54399I
u = 0.409972 + 1.042740I
a = −0.333921 + 0.008202I
b = 0.687191 + 0.474750I
−0.71901 + 1.35928I −6.71358 − 0.21049I
u = 0.409972 − 1.042740I
a = −0.333921 − 0.008202I
b = 0.687191 − 0.474750I
−0.71901 − 1.35928I −6.71358 + 0.21049I
u = −0.232752 + 1.110800I
a = −0.406380 + 1.143680I
b = −0.979109 − 0.571742I
−3.76323 − 2.26383I −13.05428 + 2.02208I
u = −0.232752 − 1.110800I
a = −0.406380 − 1.143680I
b = −0.979109 + 0.571742I
−3.76323 + 2.26383I −13.05428 − 2.02208I
u = 0.432711 + 1.187740I
a = 0.517506 + 1.268290I
b = 0.659883 − 0.866942I
−0.86443 + 6.41567I −7.45843 − 6.37638I
u = 0.432711 − 1.187740I
a = 0.517506 − 1.268290I
b = 0.659883 + 0.866942I
−0.86443 − 6.41567I −7.45843 + 6.37638I
u = 0.683039 + 0.071498I
a = −0.81507 − 1.50432I
b = 0.587913 + 0.647297I
2.38839 − 2.21658I −3.00360 + 3.59199I
u = 0.683039 − 0.071498I
a = −0.81507 + 1.50432I
b = 0.587913 − 0.647297I
2.38839 + 2.21658I −3.00360 − 3.59199I
u = 0.576371 + 1.269310I
a = −0.02214 − 1.67156I
b = −1.66528 + 0.31996I
−8.60064 + 11.02740I −8.81919 − 5.78425I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.576371 − 1.269310I
a = −0.02214 + 1.67156I
b = −1.66528 − 0.31996I
−8.60064 − 11.02740I −8.81919 + 5.78425I
u = 0.376370 + 1.343310I
a = −0.157273 − 0.643532I
b = −1.74770 − 0.06929I
−10.10760 − 0.68348I −10.33009 + 0.33748I
u = 0.376370 − 1.343310I
a = −0.157273 + 0.643532I
b = −1.74770 + 0.06929I
−10.10760 + 0.68348I −10.33009 − 0.33748I
u = −0.49655 + 1.32834I
a = 0.015741 − 1.194450I
b = 1.76275 + 0.15335I
−13.5263 − 5.3591I −12.04516 + 3.16064I
u = −0.49655 − 1.32834I
a = 0.015741 + 1.194450I
b = 1.76275 − 0.15335I
−13.5263 + 5.3591I −12.04516 − 3.16064I
u = −0.339124
a = 1.65907
b = −0.613341
−0.865956 −11.4730
7
II. I
u
2
= hb − 1, a
2
− 2au + 2a + u − 2, u
2
− u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
−u + 1
a
3
=
u
−u + 1
a
5
=
−u
u − 1
a
8
=
a
1
a
4
=
−au + u − 1
au − a + 2u − 1
a
11
=
−a + 1
−1
a
7
=
1
0
a
10
=
−a
−1
a
9
=
−u + 1
−au − 2
a
9
=
−u + 1
−au − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u − 4
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
2
− u + 1)
2
c
2
, c
5
(u
2
+ u + 1)
2
c
3
, c
4
, c
8
c
9
(u
2
+ 2)
2
c
7
(u + 1)
4
c
10
, c
11
(u − 1)
4
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
(y
2
+ y + 1)
2
c
3
, c
4
, c
8
c
9
(y + 2)
4
c
7
, c
10
, c
11
(y −1)
4
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.724745 + 0.158919I
b = 1.00000
3.28987 + 2.02988I −6.00000 − 3.46410I
u = 0.500000 + 0.866025I
a = −1.72474 + 1.57313I
b = 1.00000
3.28987 + 2.02988I −6.00000 − 3.46410I
u = 0.500000 − 0.866025I
a = 0.724745 − 0.158919I
b = 1.00000
3.28987 − 2.02988I −6.00000 + 3.46410I
u = 0.500000 − 0.866025I
a = −1.72474 − 1.57313I
b = 1.00000
3.28987 − 2.02988I −6.00000 + 3.46410I
11
III. I
u
3
= hb + 1, a − u − 1, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
3
=
−u
u + 1
a
5
=
−u
u + 1
a
8
=
u + 1
−1
a
4
=
−u
u + 1
a
11
=
u + 2
−1
a
7
=
−1
0
a
10
=
u + 1
−1
a
9
=
u + 1
−1
a
9
=
u + 1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u − 10
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u
2
+ u + 1
c
3
, c
4
, c
8
c
9
u
2
c
5
u
2
− u + 1
c
7
, c
10
(u − 1)
2
c
11
(u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
y
2
+ y + 1
c
3
, c
4
, c
8
c
9
y
2
c
7
, c
10
, c
11
(y −1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = −1.00000
−1.64493 − 2.02988I −12.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = 0.500000 − 0.866025I
b = −1.00000
−1.64493 + 2.02988I −12.00000 − 3.46410I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
2
)(u
2
+ u + 1)(u
26
− 2u
25
+ ··· − u + 3)
c
2
((u
2
+ u + 1)
3
)(u
26
+ 16u
25
+ ··· − 43u + 9)
c
3
, c
4
, c
8
u
2
(u
2
+ 2)
2
(u
26
+ u
25
+ ··· − 8u − 4)
c
5
(u
2
− u + 1)(u
2
+ u + 1)
2
(u
26
− 2u
25
+ ··· − u + 3)
c
6
((u
2
− u + 1)
2
)(u
2
+ u + 1)(u
26
+ 2u
25
+ ··· − 13u + 3)
c
7
((u − 1)
2
)(u + 1)
4
(u
26
+ 3u
25
+ ··· + 22u − 3)
c
9
u
2
(u
2
+ 2)
2
(u
26
− u
25
+ ··· − 32u − 4)
c
10
((u − 1)
6
)(u
26
+ 33u
25
+ ··· + 64u + 9)
c
11
((u − 1)
4
)(u + 1)
2
(u
26
+ 3u
25
+ ··· + 22u − 3)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
((y
2
+ y + 1)
3
)(y
26
+ 16y
25
+ ··· − 43y + 9)
c
2
((y
2
+ y + 1)
3
)(y
26
− 8y
25
+ ··· − 7123y + 81)
c
3
, c
4
, c
8
y
2
(y + 2)
4
(y
26
+ 21y
25
+ ··· + 64y + 16)
c
6
((y
2
+ y + 1)
3
)(y
26
− 32y
25
+ ··· − 187y + 9)
c
7
, c
11
((y −1)
6
)(y
26
− 33y
25
+ ··· − 64y + 9)
c
9
y
2
(y + 2)
4
(y
26
− 39y
25
+ ··· − 128y + 16)
c
10
((y −1)
6
)(y
26
− 73y
25
+ ··· + 35108y + 81)
17
