11a
4
(K11a
4
)
A knot diagram
1
Linearized knot diagam
5 1 7 2 3 10 4 11 6 9 8
Solving Sequence
1,5
2 3 6
4,8
7 11 9 10
c
1
c
2
c
5
c
4
c
7
c
11
c
8
c
10
c
3
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−6u
53
+ 25u
52
+ ··· + 4b + 7, 7u
53
− 14u
52
+ ··· + 4a + 17, u
54
− 4u
53
+ ··· − 5u + 1i
I
u
2
= h−au + b, a
3
+ a
2
u + a
2
+ 2au − 1, u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 60 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−6u
53
+25u
52
+· · ·+4b+7, 7u
53
−14u
52
+· · ·+4a+17, u
54
−4u
53
+· · ·−5u+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
8
=
−
7
4
u
53
+
7
2
u
52
+ ··· +
35
4
u −
17
4
3
2
u
53
−
25
4
u
52
+ ··· + 12u −
7
4
a
7
=
−
9
4
u
53
+
19
2
u
52
+ ··· −
43
4
u +
1
4
−
1
2
u
53
+
9
4
u
52
+ ··· + 11u −
9
4
a
11
=
−
1
4
u
53
+
3
4
u
52
+ ··· + u
2
−
5
4
u
1
4
u
53
− u
52
+ ··· +
9
4
u −
1
4
a
9
=
4u
53
− 16u
52
+ ··· + 17u −
9
2
−
5
4
u
53
+
25
4
u
52
+ ··· −
39
4
u +
5
2
a
10
=
15
4
u
53
−
55
4
u
52
+ ··· +
41
4
u − 3
−
5
4
u
53
+
29
4
u
52
+ ··· −
55
4
u +
7
2
a
10
=
15
4
u
53
−
55
4
u
52
+ ··· +
41
4
u − 3
−
5
4
u
53
+
29
4
u
52
+ ··· −
55
4
u +
7
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −5u
53
+
67
2
u
52
+ ··· − 59u + 16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
54
+ 4u
53
+ ··· + 5u + 1
c
2
u
54
+ 28u
53
+ ··· + 3u + 1
c
3
, c
7
u
54
+ u
53
+ ··· + 96u + 64
c
5
u
54
− 4u
53
+ ··· − 713u + 193
c
6
, c
9
u
54
− 3u
53
+ ··· − 2u + 1
c
8
, c
10
, c
11
u
54
− 13u
53
+ ··· + 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
54
+ 28y
53
+ ··· + 3y + 1
c
2
y
54
+ 56y
52
+ ··· + 27y + 1
c
3
, c
7
y
54
− 35y
53
+ ··· − 54272y + 4096
c
5
y
54
− 28y
53
+ ··· + 214995y + 37249
c
6
, c
9
y
54
− 13y
53
+ ··· + 4y + 1
c
8
, c
10
, c
11
y
54
+ 59y
53
+ ··· − 84y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.379892 + 0.998863I
a = 1.103900 + 0.557020I
b = 0.102602 + 0.436031I
−1.03201 − 1.50079I 0
u = −0.379892 − 0.998863I
a = 1.103900 − 0.557020I
b = 0.102602 − 0.436031I
−1.03201 + 1.50079I 0
u = −0.626212 + 0.684020I
a = −0.946934 + 0.601830I
b = 0.572857 − 0.492032I
1.52357 − 3.40391I 4.82806 + 9.13661I
u = −0.626212 − 0.684020I
a = −0.946934 − 0.601830I
b = 0.572857 + 0.492032I
1.52357 + 3.40391I 4.82806 − 9.13661I
u = 0.885029 + 0.254996I
a = −0.781886 + 1.060890I
b = 0.24815 − 1.64054I
−8.53853 − 8.83927I −0.16984 + 5.18354I
u = 0.885029 − 0.254996I
a = −0.781886 − 1.060890I
b = 0.24815 + 1.64054I
−8.53853 + 8.83927I −0.16984 − 5.18354I
u = −0.591212 + 0.904355I
a = 0.136029 − 0.965536I
b = 0.452763 + 0.325398I
0.88913 − 1.37469I 0
u = −0.591212 − 0.904355I
a = 0.136029 + 0.965536I
b = 0.452763 − 0.325398I
0.88913 + 1.37469I 0
u = 0.885260 + 0.230562I
a = 0.737527 − 0.964365I
b = −0.08283 + 1.52053I
−8.94569 − 2.41782I −0.944300 + 0.387056I
u = 0.885260 − 0.230562I
a = 0.737527 + 0.964365I
b = −0.08283 − 1.52053I
−8.94569 + 2.41782I −0.944300 − 0.387056I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.753157 + 0.805387I
a = −1.05816 + 1.16882I
b = 0.15061 − 1.52668I
−5.15771 − 5.91377I 0
u = −0.753157 − 0.805387I
a = −1.05816 − 1.16882I
b = 0.15061 + 1.52668I
−5.15771 + 5.91377I 0
u = −0.745011 + 0.831417I
a = 0.96483 − 1.25275I
b = 0.09191 + 1.50020I
−5.23505 + 0.31393I 0
u = −0.745011 − 0.831417I
a = 0.96483 + 1.25275I
b = 0.09191 − 1.50020I
−5.23505 − 0.31393I 0
u = −0.499017 + 1.041130I
a = −0.856626 − 1.113760I
b = 0.506001 − 0.649056I
−0.12306 − 4.76592I 0
u = −0.499017 − 1.041130I
a = −0.856626 + 1.113760I
b = 0.506001 + 0.649056I
−0.12306 + 4.76592I 0
u = −0.350126 + 0.758525I
a = 0.992379 − 0.143031I
b = −0.0959060 − 0.0489305I
−0.23114 − 1.44429I −1.42255 + 4.98888I
u = −0.350126 − 0.758525I
a = 0.992379 + 0.143031I
b = −0.0959060 + 0.0489305I
−0.23114 + 1.44429I −1.42255 − 4.98888I
u = 0.188104 + 0.813889I
a = 1.70541 + 0.20561I
b = 0.098016 − 1.198770I
−3.67271 − 1.63897I −3.88431 + 4.27010I
u = 0.188104 − 0.813889I
a = 1.70541 − 0.20561I
b = 0.098016 + 1.198770I
−3.67271 + 1.63897I −3.88431 − 4.27010I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.325799 + 1.118780I
a = 0.231756 + 0.233450I
b = 0.623088 − 1.028710I
−4.41304 − 1.91296I 0
u = 0.325799 − 1.118780I
a = 0.231756 − 0.233450I
b = 0.623088 + 1.028710I
−4.41304 + 1.91296I 0
u = 0.461228 + 1.103380I
a = −0.760887 + 0.869538I
b = 1.049960 − 0.133110I
−0.79930 + 3.68471I 0
u = 0.461228 − 1.103380I
a = −0.760887 − 0.869538I
b = 1.049960 + 0.133110I
−0.79930 − 3.68471I 0
u = 0.739810 + 0.258344I
a = −1.28681 + 0.82591I
b = 0.734944 − 0.813229I
−0.37627 − 5.01917I 3.74598 + 6.24423I
u = 0.739810 − 0.258344I
a = −1.28681 − 0.82591I
b = 0.734944 + 0.813229I
−0.37627 + 5.01917I 3.74598 − 6.24423I
u = 0.263878 + 0.734727I
a = −2.05477 − 0.12446I
b = 0.313156 + 1.353580I
−3.33199 + 3.97385I −2.23953 − 0.35907I
u = 0.263878 − 0.734727I
a = −2.05477 + 0.12446I
b = 0.313156 − 1.353580I
−3.33199 − 3.97385I −2.23953 + 0.35907I
u = 0.389551 + 1.166930I
a = 0.368785 − 0.096490I
b = −0.342003 + 0.642138I
−6.05420 + 2.70137I 0
u = 0.389551 − 1.166930I
a = 0.368785 + 0.096490I
b = −0.342003 − 0.642138I
−6.05420 − 2.70137I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.438089 + 1.158700I
a = 1.57252 + 1.18680I
b = 0.03942 + 1.53460I
−7.79249 − 0.92626I 0
u = −0.438089 − 1.158700I
a = 1.57252 − 1.18680I
b = 0.03942 − 1.53460I
−7.79249 + 0.92626I 0
u = −0.459497 + 1.157950I
a = −1.51732 − 1.29201I
b = 0.16035 − 1.58211I
−7.64003 − 7.27340I 0
u = −0.459497 − 1.157950I
a = −1.51732 + 1.29201I
b = 0.16035 + 1.58211I
−7.64003 + 7.27340I 0
u = 0.727484 + 0.127692I
a = 1.178260 − 0.411070I
b = −0.306147 + 0.433369I
−2.35286 − 1.07266I −1.50878 + 0.45563I
u = 0.727484 − 0.127692I
a = 1.178260 + 0.411070I
b = −0.306147 − 0.433369I
−2.35286 + 1.07266I −1.50878 − 0.45563I
u = 0.533177 + 1.145480I
a = −1.55151 + 0.68284I
b = 0.838073 + 0.873836I
−2.97092 + 9.82935I 0
u = 0.533177 − 1.145480I
a = −1.55151 − 0.68284I
b = 0.838073 − 0.873836I
−2.97092 − 9.82935I 0
u = 0.493294 + 1.165470I
a = 1.239930 − 0.400909I
b = −0.458261 − 0.419014I
−5.33044 + 5.62043I 0
u = 0.493294 − 1.165470I
a = 1.239930 + 0.400909I
b = −0.458261 + 0.419014I
−5.33044 − 5.62043I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.272316 + 1.258420I
a = 0.301237 − 0.770778I
b = 0.18966 − 1.67412I
−13.4813 − 5.1046I 0
u = 0.272316 − 1.258420I
a = 0.301237 + 0.770778I
b = 0.18966 + 1.67412I
−13.4813 + 5.1046I 0
u = 0.292425 + 1.259690I
a = −0.162036 + 0.763414I
b = −0.05529 + 1.59185I
−13.77420 + 1.43522I 0
u = 0.292425 − 1.259690I
a = −0.162036 − 0.763414I
b = −0.05529 − 1.59185I
−13.77420 − 1.43522I 0
u = 0.575070 + 1.194170I
a = −2.04207 + 0.34417I
b = 0.27962 + 1.66807I
−11.3710 + 14.1846I 0
u = 0.575070 − 1.194170I
a = −2.04207 − 0.34417I
b = 0.27962 − 1.66807I
−11.3710 − 14.1846I 0
u = 0.564569 + 1.201030I
a = 1.96539 − 0.25795I
b = −0.13428 − 1.52301I
−11.8740 + 7.7158I 0
u = 0.564569 − 1.201030I
a = 1.96539 + 0.25795I
b = −0.13428 + 1.52301I
−11.8740 − 7.7158I 0
u = −0.666189 + 0.031728I
a = −0.0922958 + 0.0482952I
b = 0.13196 + 1.49406I
−4.53071 + 3.08686I 1.77023 − 2.56143I
u = −0.666189 − 0.031728I
a = −0.0922958 − 0.0482952I
b = 0.13196 − 1.49406I
−4.53071 − 3.08686I 1.77023 + 2.56143I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.474195 + 0.404040I
a = −0.896122 + 0.464008I
b = 0.603365 + 0.428675I
1.67487 + 0.64549I 7.52215 − 2.21794I
u = −0.474195 − 0.404040I
a = −0.896122 − 0.464008I
b = 0.603365 − 0.428675I
1.67487 − 0.64549I 7.52215 + 2.21794I
u = 0.385602 + 0.286567I
a = −1.99051 + 0.52447I
b = 0.788212 + 0.173723I
1.57101 + 0.14808I 6.86906 + 0.36637I
u = 0.385602 − 0.286567I
a = −1.99051 − 0.52447I
b = 0.788212 − 0.173723I
1.57101 − 0.14808I 6.86906 − 0.36637I
10
II. I
u
2
= h−au + b, a
3
+ a
2
u + a
2
+ 2au − 1, 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
4
=
−u
u + 1
a
8
=
a
au
a
7
=
a
au
a
11
=
a
2
u + 1
−a
2
u − a
2
a
9
=
a
2
u + au − a − u − 1
−a
2
u − a
2
− au + 1
a
10
=
−a
2
− a − u
−a
2
u − a
2
− au + 1
a
10
=
−a
2
− a − u
−a
2
u − a
2
− au + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3a
2
u − 2a
2
− 3au + a + 7u + 10
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u
2
+ u + 1)
3
c
3
, c
7
u
6
c
4
(u
2
− u + 1)
3
c
6
(u
3
− u
2
+ 1)
2
c
8
(u
3
+ u
2
+ 2u + 1)
2
c
9
(u
3
+ u
2
− 1)
2
c
10
, c
11
(u
3
− u
2
+ 2u − 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)
3
c
3
, c
7
y
6
c
6
, c
9
(y
3
− y
2
+ 2y −1)
2
c
8
, c
10
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −1.239560 + 0.467306I
b = 0.215080 − 1.307140I
−3.02413 + 0.79824I 2.23639 + 1.26697I
u = −0.500000 + 0.866025I
a = 1.024480 − 0.839835I
b = 0.215080 + 1.307140I
−3.02413 − 4.85801I 0.94625 + 7.60556I
u = −0.500000 + 0.866025I
a = −0.284920 − 0.493496I
b = 0.569840
1.11345 − 2.02988I 5.31735 + 5.84990I
u = −0.500000 − 0.866025I
a = 1.024480 + 0.839835I
b = 0.215080 − 1.307140I
−3.02413 − 0.79824I 2.23639 − 1.26697I
u = −0.500000 − 0.866025I
a = −1.239560 − 0.467306I
b = 0.215080 + 1.307140I
−3.02413 + 4.85801I 0.94625 − 7.60556I
u = −0.500000 − 0.866025I
a = −0.284920 + 0.493496I
b = 0.569840
1.11345 + 2.02988I 5.31735 − 5.84990I
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
3
)(u
54
+ 4u
53
+ ··· + 5u + 1)
c
2
((u
2
+ u + 1)
3
)(u
54
+ 28u
53
+ ··· + 3u + 1)
c
3
, c
7
u
6
(u
54
+ u
53
+ ··· + 96u + 64)
c
4
((u
2
− u + 1)
3
)(u
54
+ 4u
53
+ ··· + 5u + 1)
c
5
((u
2
+ u + 1)
3
)(u
54
− 4u
53
+ ··· − 713u + 193)
c
6
((u
3
− u
2
+ 1)
2
)(u
54
− 3u
53
+ ··· − 2u + 1)
c
8
((u
3
+ u
2
+ 2u + 1)
2
)(u
54
− 13u
53
+ ··· + 4u + 1)
c
9
((u
3
+ u
2
− 1)
2
)(u
54
− 3u
53
+ ··· − 2u + 1)
c
10
, c
11
((u
3
− u
2
+ 2u − 1)
2
)(u
54
− 13u
53
+ ··· + 4u + 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
3
)(y
54
+ 28y
53
+ ··· + 3y + 1)
c
2
((y
2
+ y + 1)
3
)(y
54
+ 56y
52
+ ··· + 27y + 1)
c
3
, c
7
y
6
(y
54
− 35y
53
+ ··· − 54272y + 4096)
c
5
((y
2
+ y + 1)
3
)(y
54
− 28y
53
+ ··· + 214995y + 37249)
c
6
, c
9
((y
3
− y
2
+ 2y −1)
2
)(y
54
− 13y
53
+ ··· + 4y + 1)
c
8
, c
10
, c
11
((y
3
+ 3y
2
+ 2y −1)
2
)(y
54
+ 59y
53
+ ··· − 84y + 1)
16
