11n
9
(K11n
9
)
A knot diagram
1
Linearized knot diagam
5 1 7 2 3 10 4 11 1 7 9
Solving Sequence
1,5
2 3
6,10
7 4 8 9 11
c
1
c
2
c
5
c
6
c
4
c
7
c
9
c
11
c
3
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h7u
10
− 25u
9
+ 49u
8
− 40u
7
+ 18u
6
− 11u
5
+ 35u
4
+ u
2
+ 46b − 21u + 24,
− 31u
10
+ 114u
9
− 240u
8
+ 279u
7
− 290u
6
+ 305u
5
− 408u
4
+ 253u
3
− 241u
2
+ 46a + 116u − 241,
u
11
− 4u
10
+ 9u
9
− 12u
8
+ 13u
7
− 13u
6
+ 16u
5
− 12u
4
+ 10u
3
− 4u
2
+ 8u − 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
I
u
3
= h−au + 3b + a + u − 1, a
2
+ au − 4u − 4, u
2
+ u + 1i
* 3 irreducible components of dim
C
= 0, with total 20 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
= h7u
10
− 25u
9
+ · · · + 46b + 24, −31u
10
+ 114u
9
+ · · · + 46a −
241, u
11
− 4u
10
+ · · · + 8u − 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
10
=
0.673913u
10
− 2.47826u
9
+ ··· − 2.52174u + 5.23913
−0.152174u
10
+ 0.543478u
9
+ ··· + 0.456522u − 0.521739
a
7
=
−0.173913u
10
+ 0.978261u
9
+ ··· + 1.02174u − 3.23913
−0.282609u
10
+ 1.15217u
9
+ ··· + 1.84783u + 0.173913
a
4
=
−u
u
3
+ u
a
8
=
0.260870u
10
− 0.717391u
9
+ ··· − 1.28261u − 2.89130
−0.0217391u
10
+ 0.934783u
9
+ ··· + 4.06522u − 0.217391
a
9
=
0.521739u
10
− 1.93478u
9
+ ··· − 2.06522u + 4.71739
−0.152174u
10
+ 0.543478u
9
+ ··· + 0.456522u − 0.521739
a
11
=
0.0652174u
10
− 0.304348u
9
+ ··· − 0.695652u + 3.15217
0.108696u
10
− 0.673913u
9
+ ··· − 2.32609u + 0.0869565
a
11
=
0.0652174u
10
− 0.304348u
9
+ ··· − 0.695652u + 3.15217
0.108696u
10
− 0.673913u
9
+ ··· − 2.32609u + 0.0869565
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
13
46
u
10
+
53
46
u
9
−
137
46
u
8
+
93
23
u
7
−
89
23
u
6
+
119
46
u
5
−
203
46
u
4
+ 6u
3
−
337
46
u
2
+
39
46
u −
272
23
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
11
+ 4u
10
+ ··· + 8u + 1
c
2
u
11
+ 2u
10
+ ··· + 56u − 1
c
3
, c
7
u
11
− 3u
10
+ ··· + 16u + 16
c
5
u
11
− 4u
10
+ ··· + 790u + 97
c
6
, c
10
u
11
+ 3u
10
+ ··· − 96u + 32
c
8
, c
9
, c
11
u
11
− 8u
10
+ ··· − 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
11
+ 2y
10
+ ··· + 56y −1
c
2
y
11
+ 18y
10
+ ··· + 3376y −1
c
3
, c
7
y
11
+ 15y
10
+ ··· + 1152y −256
c
5
y
11
+ 34y
10
+ ··· + 507312y −9409
c
6
, c
10
y
11
+ 27y
10
+ ··· − 2560y −1024
c
8
, c
9
, c
11
y
11
− 14y
10
+ ··· − 114y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.370726 + 0.886061I
a = 0.869101 − 0.048452I
b = 0.0248083 + 0.1208390I
−0.37744 − 1.65887I −3.08713 + 3.12324I
u = −0.370726 − 0.886061I
a = 0.869101 + 0.048452I
b = 0.0248083 − 0.1208390I
−0.37744 + 1.65887I −3.08713 − 3.12324I
u = −0.619363 + 0.675074I
a = 1.22270 − 1.02577I
b = −1.234510 + 0.125378I
−1.43681 − 1.43186I −8.27132 + 5.43285I
u = −0.619363 − 0.675074I
a = 1.22270 + 1.02577I
b = −1.234510 − 0.125378I
−1.43681 + 1.43186I −8.27132 − 5.43285I
u = 0.684593 + 1.110730I
a = −1.60080 + 0.55358I
b = 1.67575 + 0.38496I
−8.43909 + 3.01365I −11.25510 − 3.03574I
u = 0.684593 − 1.110730I
a = −1.60080 − 0.55358I
b = 1.67575 − 0.38496I
−8.43909 − 3.01365I −11.25510 + 3.03574I
u = 0.85960 + 1.26321I
a = −0.91455 + 2.45311I
b = 1.82324 − 1.07192I
10.9219 + 10.3175I −9.91350 − 4.19094I
u = 0.85960 − 1.26321I
a = −0.91455 − 2.45311I
b = 1.82324 + 1.07192I
10.9219 − 10.3175I −9.91350 + 4.19094I
u = 1.38032 + 0.75647I
a = −2.57070 − 2.19582I
b = 1.94194 + 1.79095I
12.91330 − 2.44000I −8.55865 + 0.24092I
u = 1.38032 − 0.75647I
a = −2.57070 + 2.19582I
b = 1.94194 − 1.79095I
12.91330 + 2.44000I −8.55865 − 0.24092I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.131154
a = 4.98850
b = −0.462456
−0.844734 −11.8290
6
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
10
=
u
4
− u
3
+ 2u
2
− u + 1
−1
a
7
=
−u
4
− u
2
− 1
u
4
− u
3
+ u
2
+ 1
a
4
=
−u
u
3
+ u
a
8
=
−1
0
a
9
=
u
4
− u
3
+ 2u
2
− u
−1
a
11
=
u
4
− u
3
+ 2u
2
− u + 1
−1
a
11
=
u
4
− u
3
+ 2u
2
− u + 1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
4
+ 3u
3
− 4u
2
+ 8u − 15
7
(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
10
u
5
c
8
, c
9
(u − 1)
5
c
11
(u + 1)
5
8
(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
10
y
5
c
8
, c
9
, c
11
(y −1)
5
9
(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
−1.97403 − 1.53058I −13.5086 + 9.8710I
u = −0.339110 − 0.822375I
a = −0.428550 + 1.039280I
b = −1.00000
−1.97403 + 1.53058I −13.5086 − 9.8710I
u = 0.766826
a = 1.30408
b = −1.00000
−4.04602 −8.82740
u = 0.455697 + 1.200150I
a = 0.276511 − 0.728237I
b = −1.00000
−7.51750 + 4.40083I −11.07763 − 5.80708I
u = 0.455697 − 1.200150I
a = 0.276511 + 0.728237I
b = −1.00000
−7.51750 − 4.40083I −11.07763 + 5.80708I
10
III. I
u
3
= h−au + 3b + a + u − 1, a
2
+ au − 4u − 4, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u + 1
a
3
=
−u
u + 1
a
6
=
−1
0
a
10
=
a
1
3
au −
1
3
a −
1
3
u +
1
3
a
7
=
−
1
3
au −
2
3
a +
4
3
u +
5
3
−
1
3
au +
1
3
a +
1
3
u −
4
3
a
4
=
−u
u + 1
a
8
=
−
1
3
au −
2
3
a +
4
3
u +
5
3
−
1
3
au +
1
3
a +
1
3
u −
4
3
a
9
=
1
3
au +
2
3
a −
1
3
u +
1
3
1
3
au −
1
3
a −
1
3
u +
1
3
a
11
=
−
2
3
au −
1
3
a +
5
3
u +
7
3
−
1
3
au +
1
3
a +
1
3
u −
4
3
a
11
=
−
2
3
au −
1
3
a +
5
3
u +
7
3
−
1
3
au +
1
3
a +
1
3
u −
4
3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3au + 3a + 4u − 15
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u
2
+ u + 1)
2
c
3
, c
7
u
4
c
4
(u
2
− u + 1)
2
c
6
, c
8
, c
9
(u
2
+ u − 1)
2
c
10
, c
11
(u
2
− u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
(y
2
+ y + 1)
2
c
3
, c
7
y
4
c
6
, c
8
, c
9
c
10
, c
11
(y
2
− 3y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 1.92705 + 0.53523I
b = −0.618034
−0.98696 − 2.02988I −15.5000 + 9.2736I
u = −0.500000 + 0.866025I
a = −1.42705 − 1.40126I
b = 1.61803
−8.88264 − 2.02988I −15.5000 − 2.3454I
u = −0.500000 − 0.866025I
a = 1.92705 − 0.53523I
b = −0.618034
−0.98696 + 2.02988I −15.5000 − 9.2736I
u = −0.500000 − 0.866025I
a = −1.42705 + 1.40126I
b = 1.61803
−8.88264 + 2.02988I −15.5000 + 2.3454I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
2
)(u
5
− u
4
+ ··· + u − 1)(u
11
+ 4u
10
+ ··· + 8u + 1)
c
2
((u
2
+ u + 1)
2
)(u
5
+ 3u
4
+ ··· − u − 1)(u
11
+ 2u
10
+ ··· + 56u − 1)
c
3
u
4
(u
5
+ u
4
+ ··· + u − 1)(u
11
− 3u
10
+ ··· + 16u + 16)
c
4
((u
2
− u + 1)
2
)(u
5
+ u
4
+ ··· + u + 1)(u
11
+ 4u
10
+ ··· + 8u + 1)
c
5
((u
2
+ u + 1)
2
)(u
5
− u
4
+ ··· + u + 1)(u
11
− 4u
10
+ ··· + 790u + 97)
c
6
u
5
(u
2
+ u − 1)
2
(u
11
+ 3u
10
+ ··· − 96u + 32)
c
7
u
4
(u
5
− u
4
+ ··· + u + 1)(u
11
− 3u
10
+ ··· + 16u + 16)
c
8
, c
9
((u − 1)
5
)(u
2
+ u − 1)
2
(u
11
− 8u
10
+ ··· − 2u + 1)
c
10
u
5
(u
2
− u − 1)
2
(u
11
+ 3u
10
+ ··· − 96u + 32)
c
11
((u + 1)
5
)(u
2
− u − 1)
2
(u
11
− 8u
10
+ ··· − 2u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
2
)(y
5
+ 3y
4
+ ··· − y −1)(y
11
+ 2y
10
+ ··· + 56y −1)
c
2
(y
2
+ y + 1)
2
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
· (y
11
+ 18y
10
+ ··· + 3376y −1)
c
3
, c
7
y
4
(y
5
− 5y
4
+ ··· − y −1)(y
11
+ 15y
10
+ ··· + 1152y −256)
c
5
(y
2
+ y + 1)
2
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
· (y
11
+ 34y
10
+ ··· + 507312y −9409)
c
6
, c
10
y
5
(y
2
− 3y + 1)
2
(y
11
+ 27y
10
+ ··· − 2560y −1024)
c
8
, c
9
, c
11
((y −1)
5
)(y
2
− 3y + 1)
2
(y
11
− 14y
10
+ ··· − 114y −1)
16
