11n
15
(K11n
15
)
A knot diagram
1
Linearized knot diagam
5 1 9 2 3 10 11 3 1 7 8
Solving Sequence
7,11
8
1,3
2 10 6 5 4 9
c
7
c
11
c
2
c
10
c
6
c
5
c
4
c
9
c
1
, c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h3u
20
− 5u
19
+ ··· + 2b + 1, 4u
20
− 7u
19
+ ··· + 2a + 1, u
21
− 3u
20
+ ··· − u − 1i
I
u
2
= hau + b − a, a
2
+ au + a + u + 2, u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 25 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
= h3u
20
−5u
19
+· · ·+2b+1, 4u
20
−7u
19
+· · ·+2a+1, u
21
−3u
20
+· · ·−u−1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
u
2
a
1
=
−u
−u
3
+ u
a
3
=
−2u
20
+
7
2
u
19
+ ··· − 4u −
1
2
−
3
2
u
20
+
5
2
u
19
+ ··· −
3
2
u −
1
2
a
2
=
−
3
2
u
20
+ 2u
19
+ ··· + 2u
2
−
7
2
u
−
5
2
u
20
+ 4u
19
+ ··· −
3
2
u − 1
a
10
=
u
u
a
6
=
−u
2
+ 1
−u
2
a
5
=
1
2
u
19
− u
18
+ ··· + 3u +
1
2
1
2
u
20
−
1
2
u
19
+ ··· +
3
2
u +
1
2
a
4
=
−4u
20
+
11
2
u
19
+ ··· − 3u −
7
2
−
13
2
u
20
+
21
2
u
19
+ ··· −
15
2
u −
9
2
a
9
=
−u
5
+ 2u
3
+ u
−u
7
+ 3u
5
− 2u
3
+ u
a
9
=
−u
5
+ 2u
3
+ u
−u
7
+ 3u
5
− 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
13
2
u
20
− 11u
19
− 57u
18
+ 87u
17
+ 209u
16
−
459
2
u
15
− 462u
14
+
319
2
u
13
+
1465
2
u
12
+
258u
11
−713u
10
−
1001
2
u
9
+ 174u
8
+ 353u
7
+ 160u
6
−101u
5
+
9
2
u
4
−77u
3
+ 13u
2
−
1
2
u + 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
21
+ 3u
20
+ ··· − u − 1
c
2
u
21
+ 5u
20
+ ··· − 13u − 1
c
3
, c
8
u
21
− u
20
+ ··· + 16u + 16
c
5
u
21
− 3u
20
+ ··· − 517u − 241
c
6
, c
7
, c
10
c
11
u
21
+ 3u
20
+ ··· − u + 1
c
9
u
21
− u
20
+ ··· + 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
21
+ 5y
20
+ ··· − 13y −1
c
2
y
21
+ 25y
20
+ ··· + 31y −1
c
3
, c
8
y
21
− 25y
20
+ ··· + 1408y −256
c
5
y
21
+ 45y
20
+ ··· − 1228357y −58081
c
6
, c
7
, c
10
c
11
y
21
− 23y
20
+ ··· − y −1
c
9
y
21
+ 37y
20
+ ··· − y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.586384 + 0.784633I
a = −0.273404 − 1.248180I
b = 0.731733 + 0.893746I
9.06037 + 6.00997I −1.49297 − 4.91702I
u = −0.586384 − 0.784633I
a = −0.273404 + 1.248180I
b = 0.731733 − 0.893746I
9.06037 − 6.00997I −1.49297 + 4.91702I
u = −0.502427 + 0.810890I
a = 0.300534 + 0.922223I
b = −1.209200 − 0.664376I
9.31347 − 0.73158I −0.878702 − 0.143829I
u = −0.502427 − 0.810890I
a = 0.300534 − 0.922223I
b = −1.209200 + 0.664376I
9.31347 + 0.73158I −0.878702 + 0.143829I
u = 1.30058
a = −0.779544
b = −0.0623998
−2.53925 −3.24500
u = 0.650843 + 0.188135I
a = −0.679014 − 0.497949I
b = 0.257058 + 0.102289I
−1.259870 − 0.426532I −8.18330 + 0.83082I
u = 0.650843 − 0.188135I
a = −0.679014 + 0.497949I
b = 0.257058 − 0.102289I
−1.259870 + 0.426532I −8.18330 − 0.83082I
u = −1.349430 + 0.063463I
a = 0.17958 − 2.26263I
b = −0.08959 − 2.87750I
−3.34560 + 2.92064I −6.05745 − 2.89789I
u = −1.349430 − 0.063463I
a = 0.17958 + 2.26263I
b = −0.08959 + 2.87750I
−3.34560 − 2.92064I −6.05745 + 2.89789I
u = 1.45264 + 0.09803I
a = 0.354434 − 0.632200I
b = −0.288153 − 1.061360I
−6.20743 − 3.92323I −7.50265 + 3.86571I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.45264 − 0.09803I
a = 0.354434 + 0.632200I
b = −0.288153 + 1.061360I
−6.20743 + 3.92323I −7.50265 − 3.86571I
u = 1.51579 + 0.30268I
a = −1.36198 + 1.35966I
b = −2.52269 + 1.32132I
2.78944 − 3.34833I −3.65691 + 0.92294I
u = 1.51579 − 0.30268I
a = −1.36198 − 1.35966I
b = −2.52269 − 1.32132I
2.78944 + 3.34833I −3.65691 − 0.92294I
u = 0.033690 + 0.433118I
a = −1.030760 − 0.710959I
b = −0.333491 + 0.730890I
0.87590 − 1.40870I 1.21226 + 3.90536I
u = 0.033690 − 0.433118I
a = −1.030760 + 0.710959I
b = −0.333491 − 0.730890I
0.87590 + 1.40870I 1.21226 − 3.90536I
u = 1.56509 + 0.27721I
a = 1.22824 − 1.65327I
b = 2.69420 − 2.24461I
2.01164 − 9.94805I −4.72572 + 5.38300I
u = 1.56509 − 0.27721I
a = 1.22824 + 1.65327I
b = 2.69420 + 2.24461I
2.01164 + 9.94805I −4.72572 − 5.38300I
u = −0.317530 + 0.257874I
a = 2.02998 − 0.42048I
b = 0.636210 − 0.403748I
−0.37325 + 2.55975I 0.64804 − 6.58188I
u = −0.317530 − 0.257874I
a = 2.02998 + 0.42048I
b = 0.636210 + 0.403748I
−0.37325 − 2.55975I 0.64804 + 6.58188I
u = −1.61257 + 0.03861I
a = 0.642150 − 0.779274I
b = 1.15512 − 1.57833I
−9.12765 + 1.23257I −9.74011 + 3.00809I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.61257 − 0.03861I
a = 0.642150 + 0.779274I
b = 1.15512 + 1.57833I
−9.12765 − 1.23257I −9.74011 − 3.00809I
7
II. I
u
2
= hau + b − a, a
2
+ au + a + u + 2, u
2
+ u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u + 1
a
1
=
−u
−u + 1
a
3
=
a
−au + a
a
2
=
−au + 2a
−3au + 2a
a
10
=
u
u
a
6
=
u
u − 1
a
5
=
a + 2u + 1
−au + a + 2u − 1
a
4
=
a
−au + a
a
9
=
1
−u + 1
a
9
=
1
−u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5au + 2a + u − 8
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
, c
9
(u
2
+ u − 1)
2
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 = −0.80902 + 1.40126I
b = −0.309017 + 0.535233I
−0.98696 + 2.02988I −6.50000 − 1.52761I
u = 0.618034
a = −0.80902 − 1.40126I
b = −0.309017 − 0.535233I
−0.98696 − 2.02988I −6.50000 + 1.52761I
u = −1.61803
a = 0.309017 + 0.535233I
b = 0.80902 + 1.40126I
−8.88264 − 2.02988I −6.50000 + 5.40059I
u = −1.61803
a = 0.309017 − 0.535233I
b = 0.80902 − 1.40126I
−8.88264 + 2.02988I −6.50000 − 5.40059I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
2
)(u
21
+ 3u
20
+ ··· − u − 1)
c
2
((u
2
+ u + 1)
2
)(u
21
+ 5u
20
+ ··· − 13u − 1)
c
3
, c
8
u
4
(u
21
− u
20
+ ··· + 16u + 16)
c
4
((u
2
− u + 1)
2
)(u
21
+ 3u
20
+ ··· − u − 1)
c
5
((u
2
+ u + 1)
2
)(u
21
− 3u
20
+ ··· − 517u − 241)
c
6
, c
7
((u
2
+ u − 1)
2
)(u
21
+ 3u
20
+ ··· − u + 1)
c
9
((u
2
+ u − 1)
2
)(u
21
− u
20
+ ··· + 3u + 1)
c
10
, c
11
((u
2
− u − 1)
2
)(u
21
+ 3u
20
+ ··· − u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
2
)(y
21
+ 5y
20
+ ··· − 13y −1)
c
2
((y
2
+ y + 1)
2
)(y
21
+ 25y
20
+ ··· + 31y −1)
c
3
, c
8
y
4
(y
21
− 25y
20
+ ··· + 1408y −256)
c
5
((y
2
+ y + 1)
2
)(y
21
+ 45y
20
+ ··· − 1228357y −58081)
c
6
, c
7
, c
10
c
11
((y
2
− 3y + 1)
2
)(y
21
− 23y
20
+ ··· − y −1)
c
9
((y
2
− 3y + 1)
2
)(y
21
+ 37y
20
+ ··· − y −1)
13
