11n
14
(K11n
14
)
A knot diagram
1
Linearized knot diagam
5 1 9 2 3 10 11 3 6 7 9
Solving Sequence
6,10
7
3,11
5 9 4 1 2 8
c
6
c
10
c
5
c
9
c
3
c
11
c
2
c
8
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−3u
25
− 6u
24
+ ··· + 2b − 7u, 7u
25
+ 14u
24
+ ··· + 2a + 9u, u
26
+ 3u
25
+ ··· + u − 1i
I
u
2
= hb + a, a
2
− a + 1, u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 30 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−3u
25
−6u
24
+· · ·+ 2 b−7u, 7u
25
+14u
24
+· · ·+ 2 a+ 9u, u
26
+3u
25
+· · ·+u−1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
−u
2
a
3
=
−
7
2
u
25
− 7u
24
+ ··· − 2u
2
−
9
2
u
3
2
u
25
+ 3u
24
+ ··· + u
2
+
7
2
u
a
11
=
u
−u
3
+ u
a
5
=
1
2
u
25
+ u
24
+ ··· − 4u
2
−
5
2
u
−
1
2
u
25
− u
24
+ ··· + 4u
2
+
1
2
u
a
9
=
−u
u
a
4
=
−
13
2
u
25
− 12u
24
+ ··· −
17
2
u + 2
9
2
u
25
+ 8u
24
+ ··· +
15
2
u − 2
a
1
=
−u
5
+ 2u
3
+ u
u
5
− 3u
3
+ u
a
2
=
−6u
25
− 10u
24
+ ··· − 6u + 2
11
2
u
25
+ 9u
24
+ ··· +
15
2
u − 2
a
8
=
−u
2
+ 1
u
4
− 2u
2
a
8
=
−u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
5
2
u
25
+ 4u
24
−
57
2
u
23
− 41u
22
+
277
2
u
21
+
305
2
u
20
− 391u
19
−
205u
18
+
1475
2
u
17
− 141u
16
− 910u
15
+ 742u
14
+ 464u
13
−
1721
2
u
12
+ 374u
11
+ 381u
10
−
512u
9
+ 147u
8
+
229
2
u
7
− 247u
6
−
49
2
u
5
+
29
2
u
4
− 53u
3
− 27u
2
−
29
2
u + 9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
26
+ 3u
25
+ ··· − 3u + 1
c
2
u
26
+ 15u
25
+ ··· − 23u + 1
c
3
, c
8
u
26
− u
25
+ ··· + 16u − 16
c
5
u
26
− 3u
25
+ ··· − 11u + 2
c
6
, c
7
, c
9
c
10
u
26
− 3u
25
+ ··· − u − 1
c
11
u
26
+ 3u
25
+ ··· + 3u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
26
+ 15y
25
+ ··· − 23y + 1
c
2
y
26
− 5y
25
+ ··· − 795y + 1
c
3
, c
8
y
26
+ 25y
25
+ ··· + 1664y + 256
c
5
y
26
− 25y
25
+ ··· + 7y + 4
c
6
, c
7
, c
9
c
10
y
26
− 29y
25
+ ··· − 19y + 1
c
11
y
26
+ 31y
25
+ ··· − 19y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.608473 + 0.715807I
a = 0.153590 − 1.045830I
b = −1.73126 + 0.24397I
−7.73283 + 7.28919I 4.65018 − 5.96812I
u = 0.608473 − 0.715807I
a = 0.153590 + 1.045830I
b = −1.73126 − 0.24397I
−7.73283 − 7.28919I 4.65018 + 5.96812I
u = 0.433445 + 0.761836I
a = −0.011324 − 1.102790I
b = −1.56800 + 0.06124I
−8.25588 − 2.37235I 3.41364 + 0.56644I
u = 0.433445 − 0.761836I
a = −0.011324 + 1.102790I
b = −1.56800 − 0.06124I
−8.25588 + 2.37235I 3.41364 − 0.56644I
u = 0.514434 + 0.670493I
a = −0.106433 + 1.146570I
b = 1.59480 − 0.23303I
−4.15442 + 2.25820I 7.09524 − 3.00458I
u = 0.514434 − 0.670493I
a = −0.106433 − 1.146570I
b = 1.59480 + 0.23303I
−4.15442 − 2.25820I 7.09524 + 3.00458I
u = −0.730522 + 0.264601I
a = −0.408112 − 0.721539I
b = −0.020892 − 0.242346I
0.141642 − 0.491245I 7.19488 + 1.21216I
u = −0.730522 − 0.264601I
a = −0.408112 + 0.721539I
b = −0.020892 + 0.242346I
0.141642 + 0.491245I 7.19488 − 1.21216I
u = 1.42824 + 0.09847I
a = −0.332202 + 0.112416I
b = 1.191590 − 0.262632I
3.94867 + 3.99401I 9.09163 − 3.57778I
u = 1.42824 − 0.09847I
a = −0.332202 − 0.112416I
b = 1.191590 + 0.262632I
3.94867 − 3.99401I 9.09163 + 3.57778I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.47226 + 0.03460I
a = −0.29209 − 2.37542I
b = 0.36258 + 1.62911I
6.47513 − 2.78553I 10.00226 + 3.18308I
u = −1.47226 − 0.03460I
a = −0.29209 + 2.37542I
b = 0.36258 − 1.62911I
6.47513 + 2.78553I 10.00226 − 3.18308I
u = −1.45262 + 0.27035I
a = 0.87552 + 1.41378I
b = −1.239850 − 0.375242I
−2.20853 − 1.36342I 6.29553 + 0.38377I
u = −1.45262 − 0.27035I
a = 0.87552 − 1.41378I
b = −1.239850 + 0.375242I
−2.20853 + 1.36342I 6.29553 − 0.38377I
u = −0.230011 + 0.458848I
a = 0.727275 + 1.025970I
b = 0.536127 + 0.217517I
−1.40190 − 2.19157I 3.35211 + 5.42014I
u = −0.230011 − 0.458848I
a = 0.727275 − 1.025970I
b = 0.536127 − 0.217517I
−1.40190 + 2.19157I 3.35211 − 5.42014I
u = 1.49087
a = 0.257464
b = −0.956110
7.14521 13.5410
u = −1.52539 + 0.21566I
a = −1.11758 − 1.58851I
b = 1.54625 + 0.70491I
2.54423 − 5.47373I 10.67253 + 2.88121I
u = −1.52539 − 0.21566I
a = −1.11758 + 1.58851I
b = 1.54625 − 0.70491I
2.54423 + 5.47373I 10.67253 − 2.88121I
u = −0.448296
a = −0.791985
b = −0.195879
0.706372 14.0850
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.56905 + 0.24097I
a = 1.25395 + 1.44166I
b = −1.78957 − 0.55743I
−0.54439 − 10.83970I 8.04696 + 6.04188I
u = −1.56905 − 0.24097I
a = 1.25395 − 1.44166I
b = −1.78957 + 0.55743I
−0.54439 + 10.83970I 8.04696 − 6.04188I
u = 1.63847 + 0.03227I
a = 0.0580399 − 0.1115770I
b = −0.266246 + 0.474227I
8.45818 + 1.37920I 7.00000 + 2.69707I
u = 1.63847 − 0.03227I
a = 0.0580399 + 0.1115770I
b = −0.266246 − 0.474227I
8.45818 − 1.37920I 7.00000 − 2.69707I
u = 0.335499 + 0.109869I
a = −0.03337 + 2.42886I
b = 0.460465 − 1.052830I
0.44924 + 2.24817I 0.24032 − 5.78182I
u = 0.335499 − 0.109869I
a = −0.03337 − 2.42886I
b = 0.460465 + 1.052830I
0.44924 − 2.24817I 0.24032 + 5.78182I
7
II. I
u
2
= hb + a, a
2
− a + 1, u
2
− u − 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
−u − 1
a
3
=
a
−a
a
11
=
u
−u − 1
a
5
=
a
−a + 1
a
9
=
−u
u
a
4
=
a
−a
a
1
=
−1
0
a
2
=
0
−a
a
8
=
−u
u
a
8
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2au + 3a − u + 12
8
(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
8
u
4
c
4
(u
2
− u + 1)
2
c
6
, c
7
(u
2
− u − 1)
2
c
9
, c
10
, c
11
(u
2
+ u − 1)
2
9
(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
8
y
4
c
6
, c
7
, c
9
c
10
, c
11
(y
2
− 3y + 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 0.500000 + 0.866025I
b = −0.500000 − 0.866025I
0.98696 − 2.02988I 13.50000 + 1.52761I
u = −0.618034
a = 0.500000 − 0.866025I
b = −0.500000 + 0.866025I
0.98696 + 2.02988I 13.50000 − 1.52761I
u = 1.61803
a = 0.500000 + 0.866025I
b = −0.500000 − 0.866025I
8.88264 − 2.02988I 13.5000 + 5.4006I
u = 1.61803
a = 0.500000 − 0.866025I
b = −0.500000 + 0.866025I
8.88264 + 2.02988I 13.5000 − 5.4006I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
2
)(u
26
+ 3u
25
+ ··· − 3u + 1)
c
2
((u
2
+ u + 1)
2
)(u
26
+ 15u
25
+ ··· − 23u + 1)
c
3
, c
8
u
4
(u
26
− u
25
+ ··· + 16u − 16)
c
4
((u
2
− u + 1)
2
)(u
26
+ 3u
25
+ ··· − 3u + 1)
c
5
((u
2
+ u + 1)
2
)(u
26
− 3u
25
+ ··· − 11u + 2)
c
6
, c
7
((u
2
− u − 1)
2
)(u
26
− 3u
25
+ ··· − u − 1)
c
9
, c
10
((u
2
+ u − 1)
2
)(u
26
− 3u
25
+ ··· − u − 1)
c
11
((u
2
+ u − 1)
2
)(u
26
+ 3u
25
+ ··· + 3u − 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
2
)(y
26
+ 15y
25
+ ··· − 23y + 1)
c
2
((y
2
+ y + 1)
2
)(y
26
− 5y
25
+ ··· − 795y + 1)
c
3
, c
8
y
4
(y
26
+ 25y
25
+ ··· + 1664y + 256)
c
5
((y
2
+ y + 1)
2
)(y
26
− 25y
25
+ ··· + 7y + 4)
c
6
, c
7
, c
9
c
10
((y
2
− 3y + 1)
2
)(y
26
− 29y
25
+ ··· − 19y + 1)
c
11
((y
2
− 3y + 1)
2
)(y
26
+ 31y
25
+ ··· − 19y + 1)
13
