11n
63
(K11n
63
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 11 10 4 5 7 6 9
Solving Sequence
7,9 4,10
3 6 11 1 2 5 8
c
9
c
3
c
6
c
10
c
11
c
2
c
5
c
8
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
14
− 8u
12
− 23u
10
− 28u
8
− 14u
6
+ 2u
5
− 4u
4
+ 6u
3
+ u
2
+ b + 2u, −u
22
+ u
21
+ ··· + a − 1,
u
23
− 2u
22
+ ··· + 12u
3
+ 1i
I
u
2
= h−u
3
− u
2
+ b − 2u − 1, a, u
4
+ u
3
+ 3u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 27 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−u
14
−8u
12
+· · ·+b +2u, −u
22
+u
21
+· · ·+a −1, u
23
−2u
22
+· · ·+12u
3
+1i
(i) Arc colorings
a
7
=
0
u
a
9
=
1
0
a
4
=
u
22
− u
21
+ ··· + 4u
2
+ 1
u
14
+ 8u
12
+ 23u
10
+ 28u
8
+ 14u
6
− 2u
5
+ 4u
4
− 6u
3
− u
2
− 2u
a
10
=
1
u
2
a
3
=
u
22
− u
21
+ ··· + 4u
2
+ 1
u
21
− 2u
20
+ ··· − u + 1
a
6
=
u
u
3
+ u
a
11
=
u
2
+ 1
u
4
+ 2u
2
a
1
=
u
4
+ 3u
2
+ 1
u
4
+ 2u
2
a
2
=
−u
13
− 8u
11
− 23u
9
− 28u
7
− 14u
5
+ 2u
4
− 4u
3
+ 6u
2
+ u + 2
−u
22
+ 2u
21
+ ··· − u + 1
a
5
=
u
3
+ 2u
u
5
+ 3u
3
+ u
a
8
=
u
8
+ 5u
6
+ 7u
4
+ 2u
2
+ 1
u
10
+ 6u
8
+ 11u
6
+ 6u
4
+ u
2
a
8
=
u
8
+ 5u
6
+ 7u
4
+ 2u
2
+ 1
u
10
+ 6u
8
+ 11u
6
+ 6u
4
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
22
+ 2u
21
− 16u
20
+ 27u
19
− 108u
18
+ 155u
17
− 403u
16
+
495u
15
− 916u
14
+ 969u
13
− 1323u
12
+ 1215u
11
− 1240u
10
+ 1001u
9
− 769u
8
+ 558u
7
−
333u
6
+ 218u
5
− 114u
4
+ 53u
3
− 24u
2
+ 3u − 5
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
23
− 5u
22
+ ··· − 4u + 1
c
2
u
23
+ 5u
22
+ ··· + 14u + 1
c
3
, c
7
u
23
+ u
22
+ ··· + 24u + 16
c
5
, c
6
, c
9
c
10
u
23
− 2u
22
+ ··· + 12u
3
+ 1
c
8
u
23
− 2u
22
+ ··· + 2u + 1
c
11
u
23
+ 8u
22
+ ··· + 168u + 49
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
23
− 5y
22
+ ··· + 14y −1
c
2
y
23
+ 31y
22
+ ··· + 14y −1
c
3
, c
7
y
23
+ 27y
22
+ ··· − 2240y −256
c
5
, c
6
, c
9
c
10
y
23
+ 28y
22
+ ··· + 60y
2
− 1
c
8
y
23
− 28y
22
+ ··· − 40y
2
− 1
c
11
y
23
− 16y
22
+ ··· + 77224y −2401
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.430869 + 0.879813I
a = −1.53457 − 0.83314I
b = −0.032165 − 1.283100I
7.83203 − 0.28979I −2.09012 + 1.34957I
u = 0.430869 − 0.879813I
a = −1.53457 + 0.83314I
b = −0.032165 + 1.283100I
7.83203 + 0.28979I −2.09012 − 1.34957I
u = 0.506895 + 0.810130I
a = 1.71344 + 0.53833I
b = 0.75596 + 1.83026I
7.20945 − 7.39071I −3.32519 + 6.20381I
u = 0.506895 − 0.810130I
a = 1.71344 − 0.53833I
b = 0.75596 − 1.83026I
7.20945 + 7.39071I −3.32519 − 6.20381I
u = −0.257149 + 0.694856I
a = −0.700814 − 0.988990I
b = −0.828985 + 0.500798I
0.92312 + 1.99790I −2.34638 − 5.92992I
u = −0.257149 − 0.694856I
a = −0.700814 + 0.988990I
b = −0.828985 − 0.500798I
0.92312 − 1.99790I −2.34638 + 5.92992I
u = −0.474423 + 0.490062I
a = 0.599724 − 0.678621I
b = −0.223191 − 0.754283I
−0.68293 + 1.66090I −3.45266 − 4.83485I
u = −0.474423 − 0.490062I
a = 0.599724 + 0.678621I
b = −0.223191 + 0.754283I
−0.68293 − 1.66090I −3.45266 + 4.83485I
u = 0.679084 + 0.057677I
a = −0.18571 + 2.39295I
b = −0.49465 + 1.44023I
4.95941 + 3.41645I −6.52166 − 2.22573I
u = 0.679084 − 0.057677I
a = −0.18571 − 2.39295I
b = −0.49465 − 1.44023I
4.95941 − 3.41645I −6.52166 + 2.22573I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.167283 + 0.490089I
a = 0.742425 + 0.936392I
b = 0.580042 − 0.782556I
−1.35992 − 0.76790I −3.34761 − 1.39618I
u = 0.167283 − 0.490089I
a = 0.742425 − 0.936392I
b = 0.580042 + 0.782556I
−1.35992 + 0.76790I −3.34761 + 1.39618I
u = −0.12264 + 1.52753I
a = −0.474160 + 0.043990I
b = 0.631841 + 0.749566I
6.06133 + 3.74831I 0.03467 − 4.58469I
u = −0.12264 − 1.52753I
a = −0.474160 − 0.043990I
b = 0.631841 − 0.749566I
6.06133 − 3.74831I 0.03467 + 4.58469I
u = 0.02397 + 1.58265I
a = −0.500609 − 0.414161I
b = −1.26560 + 1.17058I
5.91283 − 1.29853I −3.45106 + 0.05233I
u = 0.02397 − 1.58265I
a = −0.500609 + 0.414161I
b = −1.26560 − 1.17058I
5.91283 + 1.29853I −3.45106 − 0.05233I
u = −0.06369 + 1.61667I
a = 0.281195 + 0.690174I
b = 0.861442 − 0.947874I
8.92549 + 3.15334I −1.05029 − 3.26062I
u = −0.06369 − 1.61667I
a = 0.281195 − 0.690174I
b = 0.861442 + 0.947874I
8.92549 − 3.15334I −1.05029 + 3.26062I
u = 0.14770 + 1.64559I
a = −1.073510 + 0.242410I
b = −0.89607 − 2.20609I
15.6090 − 9.9000I −1.58759 + 4.90312I
u = 0.14770 − 1.64559I
a = −1.073510 − 0.242410I
b = −0.89607 + 2.20609I
15.6090 + 9.9000I −1.58759 − 4.90312I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.11544 + 1.66378I
a = 1.121440 + 0.050156I
b = 0.53774 + 1.34352I
16.6150 − 2.3845I −0.450202 + 0.532296I
u = 0.11544 − 1.66378I
a = 1.121440 − 0.050156I
b = 0.53774 − 1.34352I
16.6150 + 2.3845I −0.450202 − 0.532296I
u = −0.306699
a = 2.02230
b = 0.747269
−0.900453 −11.8240
7
II. I
u
2
= h−u
3
− u
2
+ b − 2u − 1, a, u
4
+ u
3
+ 3u
2
+ 2u + 1i
(i) Arc colorings
a
7
=
0
u
a
9
=
1
0
a
4
=
0
u
3
+ u
2
+ 2u + 1
a
10
=
1
u
2
a
3
=
0
u
3
+ u
2
+ 2u + 1
a
6
=
u
u
3
+ u
a
11
=
u
2
+ 1
−u
3
− u
2
− 2u − 1
a
1
=
−u
3
− 2u
−u
3
− u
2
− 2u − 1
a
2
=
−u
3
− 2u
0
a
5
=
u
3
+ 2u
u
3
+ u
2
+ 2u + 1
a
8
=
0
u
a
8
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5u
3
− 5u
2
− 14u − 16
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
4
c
2
, c
4
(u + 1)
4
c
3
, c
7
u
4
c
5
, c
6
u
4
− u
3
+ 3u
2
− 2u + 1
c
8
, c
11
u
4
+ u
3
+ u
2
+ 1
c
9
, c
10
u
4
+ u
3
+ 3u
2
+ 2u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
7
y
4
c
5
, c
6
, c
9
c
10
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
8
, c
11
y
4
+ y
3
+ 3y
2
+ 2y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.395123 + 0.506844I
a = 0
b = 0.351808 + 0.720342I
−1.85594 + 1.41510I −11.17855 − 5.62908I
u = −0.395123 − 0.506844I
a = 0
b = 0.351808 − 0.720342I
−1.85594 − 1.41510I −11.17855 + 5.62908I
u = −0.10488 + 1.55249I
a = 0
b = −0.851808 − 0.911292I
5.14581 + 3.16396I −6.32145 − 1.65351I
u = −0.10488 − 1.55249I
a = 0
b = −0.851808 + 0.911292I
5.14581 − 3.16396I −6.32145 + 1.65351I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
4
)(u
23
− 5u
22
+ ··· − 4u + 1)
c
2
((u + 1)
4
)(u
23
+ 5u
22
+ ··· + 14u + 1)
c
3
, c
7
u
4
(u
23
+ u
22
+ ··· + 24u + 16)
c
4
((u + 1)
4
)(u
23
− 5u
22
+ ··· − 4u + 1)
c
5
, c
6
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
23
− 2u
22
+ ··· + 12u
3
+ 1)
c
8
(u
4
+ u
3
+ u
2
+ 1)(u
23
− 2u
22
+ ··· + 2u + 1)
c
9
, c
10
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
23
− 2u
22
+ ··· + 12u
3
+ 1)
c
11
(u
4
+ u
3
+ u
2
+ 1)(u
23
+ 8u
22
+ ··· + 168u + 49)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
4
)(y
23
− 5y
22
+ ··· + 14y −1)
c
2
((y −1)
4
)(y
23
+ 31y
22
+ ··· + 14y −1)
c
3
, c
7
y
4
(y
23
+ 27y
22
+ ··· − 2240y −256)
c
5
, c
6
, c
9
c
10
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
23
+ 28y
22
+ ··· + 60y
2
− 1)
c
8
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
23
− 28y
22
+ ··· − 40y
2
− 1)
c
11
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
23
− 16y
22
+ ··· + 77224y −2401)
13
