11n
65
(K11n
65
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 9 11 3 6 5 1 6
Solving Sequence
2,4 5,9
6 10 1 8 3 11 7
c
4
c
5
c
9
c
1
c
8
c
3
c
11
c
6
c
2
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h2666u
12
− 2811u
11
+ ··· + 9382b + 3250, −1563u
12
+ 2980u
11
+ ··· + 18764a + 7547,
u
13
− 2u
12
− 2u
11
+ 7u
10
− u
9
− 9u
8
+ 16u
7
− 18u
6
− 2u
5
+ 33u
4
− 24u
3
− 10u
2
+ 17u − 4i
I
u
2
= h−u
4
+ u
3
+ 2u
2
+ b − a − u − 1, u
4
− 2u
2
a − 2u
3
+ a
2
+ au + 2a + u + 2, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
I
u
3
= h−u
2
a + au − u
2
+ b + u − 1, 2u
2
a + a
2
− 4au + 3u
2
+ 2a − 6u + 5, u
3
− u
2
+ 1i
I
u
4
= h2b + 1, 2a − 1, u + 1i
* 4 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
= h2666u
12
− 2811u
11
+ · · · + 9382b + 3250, −1563u
12
+ 2980u
11
+
· · · + 18764a + 7547, u
13
− 2u
12
+ · · · + 17u − 4i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
9
=
0.0832978u
12
− 0.158815u
11
+ ··· − 2.36613u − 0.402206
−0.284161u
12
+ 0.299616u
11
+ ··· + 3.21925u − 0.346408
a
6
=
−0.0329887u
12
+ 0.0405031u
11
+ ··· + 0.801428u + 1.42640
0.0613942u
12
− 0.0681091u
11
+ ··· − 0.833191u + 0.0125773
a
10
=
0.00314432u
12
+ 0.0551055u
11
+ ··· + 0.652206u − 0.779738
−0.0254743u
12
+ 0.129823u
11
+ ··· + 1.98721u − 0.131955
a
1
=
u
u
a
8
=
−0.0853763u
12
− 0.131848u
11
+ ··· − 2.30228u − 1.25187
−0.481454u
12
+ 0.401300u
11
+ ··· + 4.61810u − 0.548284
a
3
=
−u
3
−u
3
+ u
a
11
=
−0.168674u
12
+ 0.0269665u
11
+ ··· + 1.06385u − 0.849659
−0.197293u
12
+ 0.101684u
11
+ ··· + 2.39885u − 0.201876
a
7
=
0.334683u
12
− 0.465039u
11
+ ··· − 4.08719u + 3.28384
0.169154u
12
− 0.382967u
11
+ ··· − 3.37114u + 0.654445
a
7
=
0.334683u
12
− 0.465039u
11
+ ··· − 4.08719u + 3.28384
0.169154u
12
− 0.382967u
11
+ ··· − 3.37114u + 0.654445
(ii) Obstruction class = −1
(iii) Cusp Shapes =
34825
18764
u
12
−
68759
18764
u
11
+ ··· −
496503
18764
u +
97945
4691
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
13
− 2u
12
+ ··· + 17u − 4
c
2
u
13
+ 8u
12
+ ··· + 209u + 16
c
3
, c
7
u
13
+ 3u
12
+ ··· − 2u − 8
c
5
, c
6
, c
8
c
9
, c
11
u
13
− u
12
+ ··· + u − 1
c
10
u
13
+ 19u
12
+ ··· − 13u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
13
− 8y
12
+ ··· + 209y −16
c
2
y
13
− 4y
12
+ ··· + 22817y −256
c
3
, c
7
y
13
− 3y
12
+ ··· + 180y −64
c
5
, c
6
, c
8
c
9
, c
11
y
13
+ 19y
12
+ ··· − 13y −1
c
10
y
13
− 49y
12
+ ··· − 61y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.955186 + 0.433947I
a = −0.471198 − 0.436221I
b = 0.755123 − 0.031476I
0.903643 − 0.585016I 4.40140 + 1.35233I
u = 0.955186 − 0.433947I
a = −0.471198 + 0.436221I
b = 0.755123 + 0.031476I
0.903643 + 0.585016I 4.40140 − 1.35233I
u = −0.933504 + 0.177892I
a = −0.617558 + 0.567691I
b = −0.473971 + 0.403774I
−1.65953 + 0.62739I −3.40176 − 1.52650I
u = −0.933504 − 0.177892I
a = −0.617558 − 0.567691I
b = −0.473971 − 0.403774I
−1.65953 − 0.62739I −3.40176 + 1.52650I
u = 0.869334 + 0.624757I
a = 0.420041 + 0.925939I
b = −0.581132 + 0.140274I
1.00399 − 3.84064I 5.54977 + 8.01840I
u = 0.869334 − 0.624757I
a = 0.420041 − 0.925939I
b = −0.581132 − 0.140274I
1.00399 + 3.84064I 5.54977 − 8.01840I
u = −0.028967 + 1.273930I
a = −0.158932 + 0.197003I
b = 0.05120 − 2.05742I
−12.48260 + 4.81706I −0.19074 − 2.27482I
u = −0.028967 − 1.273930I
a = −0.158932 − 0.197003I
b = 0.05120 + 2.05742I
−12.48260 − 4.81706I −0.19074 + 2.27482I
u = 1.46956 + 0.59251I
a = −0.85770 − 1.47458I
b = 0.52077 − 2.38909I
−17.2166 − 11.4167I −1.78764 + 5.02800I
u = 1.46956 − 0.59251I
a = −0.85770 + 1.47458I
b = 0.52077 + 2.38909I
−17.2166 + 11.4167I −1.78764 − 5.02800I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.49484 + 0.63529I
a = 0.82284 − 1.22627I
b = −0.74959 − 1.95511I
−17.0497 + 2.0233I −2.18781 − 0.87077I
u = −1.49484 − 0.63529I
a = 0.82284 + 1.22627I
b = −0.74959 + 1.95511I
−17.0497 − 2.0233I −2.18781 + 0.87077I
u = 0.326480
a = −1.02499
b = 0.455205
0.885241 11.4840
6
II. I
u
2
= h−u
4
+ u
3
+ 2u
2
+ b − a − u − 1, u
4
− 2u
2
a − 2u
3
+ a
2
+ au + 2a +
u + 2, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
9
=
a
u
4
− u
3
− 2u
2
+ a + u + 1
a
6
=
u
4
a − 2u
2
a − 2u
2
+ a + u + 2
u
3
a − u
4
+ 2u
3
− 2au − 2u + 1
a
10
=
u
4
− u
2
a − u
3
− 2u
2
+ 2a + u + 1
−u
4
a + u
4
+ u
2
a − u
3
− 2u
2
+ a + 1
a
1
=
u
u
a
8
=
−2u
4
+ 3u
2
+ 1
−u
4
+ 2u
2
a
3
=
−u
3
−u
3
+ u
a
11
=
u
4
a + u
3
a + u
4
− 2u
2
a − 2au − 2u
2
+ a + u + 1
u
3
a + u
4
− 2au − 2u
2
+ 1
a
7
=
4u
4
− 6u
2
− 2u − 2
2u
4
− 4u
2
− u
a
7
=
4u
4
− 6u
2
− 2u − 2
2u
4
− 4u
2
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
3
− 8u − 2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
c
2
(u
5
+ 5u
4
+ 8u
3
+ 3u
2
− u + 1)
2
c
3
, c
7
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
c
5
, c
6
, c
8
c
9
, c
11
u
10
+ 3u
9
+ ··· + 32u + 17
c
10
u
10
+ 11u
9
+ ··· + 1016u + 289
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
2
c
2
(y
5
− 9y
4
+ 32y
3
− 35y
2
− 5y − 1)
2
c
3
, c
7
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
2
c
5
, c
6
, c
8
c
9
, c
11
y
10
+ 11y
9
+ ··· + 1016y + 289
c
10
y
10
− 25y
9
+ ··· + 78660y + 83521
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.21774
a = 1.09175 + 2.32396I
b = 1.91295 + 2.32396I
−5.69095 0.518860
u = −1.21774
a = 1.09175 − 2.32396I
b = 1.91295 − 2.32396I
−5.69095 0.518860
u = −0.309916 + 0.549911I
a = −0.653120 + 0.123189I
b = 0.12468 + 1.50332I
−3.61897 + 1.53058I 1.48489 − 4.43065I
u = −0.309916 + 0.549911I
a = −1.44967 − 1.35480I
b = −0.671868 + 0.025324I
−3.61897 + 1.53058I 1.48489 − 4.43065I
u = −0.309916 − 0.549911I
a = −0.653120 − 0.123189I
b = 0.12468 − 1.50332I
−3.61897 − 1.53058I 1.48489 + 4.43065I
u = −0.309916 − 0.549911I
a = −1.44967 + 1.35480I
b = −0.671868 − 0.025324I
−3.61897 − 1.53058I 1.48489 + 4.43065I
u = 1.41878 + 0.21917I
a = 0.171660 − 0.827142I
b = −0.516743 − 0.720802I
−9.16243 − 4.40083I −2.74431 + 3.49859I
u = 1.41878 + 0.21917I
a = 0.33938 + 1.85177I
b = −0.34902 + 1.95811I
−9.16243 − 4.40083I −2.74431 + 3.49859I
u = 1.41878 − 0.21917I
a = 0.171660 + 0.827142I
b = −0.516743 + 0.720802I
−9.16243 + 4.40083I −2.74431 − 3.49859I
u = 1.41878 − 0.21917I
a = 0.33938 − 1.85177I
b = −0.34902 − 1.95811I
−9.16243 + 4.40083I −2.74431 − 3.49859I
10
III.
I
u
3
= h−u
2
a+au−u
2
+b+u−1, 2u
2
a+a
2
−4au+3u
2
+2a−6u+5, u
3
−u
2
+1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
u
2
a
9
=
a
u
2
a − au + u
2
− u + 1
a
6
=
u
2
a − au + 3u
2
+ a − 5u + 4
u
2
a − au + 2u
2
− 3u + 3
a
10
=
−au + u
2
+ a − u + 1
−au + 2u
2
+ a − 2u + 1
a
1
=
u
u
a
8
=
au − u
2
+ u − 1
u
2
a − u
2
− a + u
a
3
=
−u
2
+ 1
−u
2
+ u + 1
a
11
=
−au + u
2
+ a + 1
−au + 2u
2
+ a − u + 1
a
7
=
−u
2
a + u − 1
−u
2
a + u − 1
a
7
=
−u
2
a + u − 1
−u
2
a + u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u − 4
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
3
+ u
2
− 1)
2
c
2
(u
3
+ u
2
+ 2u + 1)
2
c
3
, c
7
u
6
− 3u
4
+ 2u
2
+ 1
c
4
(u
3
− u
2
+ 1)
2
c
5
, c
6
, c
8
c
9
, c
11
(u
2
+ 1)
3
c
10
(u − 1)
6
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
3
− y
2
+ 2y − 1)
2
c
2
(y
3
+ 3y
2
+ 2y − 1)
2
c
3
, c
7
(y
3
− 3y
2
+ 2y + 1)
2
c
5
, c
6
, c
8
c
9
, c
11
(y + 1)
6
c
10
(y − 1)
6
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 1.102080 + 0.844941I
b = −0.867423 + 0.622301I
−0.26574 − 2.82812I −0.49024 + 2.97945I
u = 0.877439 + 0.744862I
a = −0.022482 − 0.479777I
b = 0.622301 + 0.867423I
−0.26574 − 2.82812I −0.49024 + 2.97945I
u = 0.877439 − 0.744862I
a = 1.102080 − 0.844941I
b = −0.867423 − 0.622301I
−0.26574 + 2.82812I −0.49024 − 2.97945I
u = 0.877439 − 0.744862I
a = −0.022482 + 0.479777I
b = 0.622301 − 0.867423I
−0.26574 + 2.82812I −0.49024 − 2.97945I
u = −0.754878
a = −3.07960 + 1.32472I
b = −1.75488 + 1.75488I
−4.40332 −7.01950
u = −0.754878
a = −3.07960 − 1.32472I
b = −1.75488 − 1.75488I
−4.40332 −7.01950
14
IV. I
u
4
= h2b + 1, 2a − 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
4
=
1
0
a
5
=
1
1
a
9
=
0.5
−0.5
a
6
=
1.5
0.5
a
10
=
−0.5
−1.5
a
1
=
−1
−1
a
8
=
2
0
a
3
=
1
0
a
11
=
0.5
−0.5
a
7
=
2
0
a
7
=
2
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2.25
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
, c
9
c
11
u − 1
c
2
, c
4
, c
5
c
6
, c
10
u + 1
c
3
, c
7
u
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
8
c
9
, c
10
, c
11
y − 1
c
3
, c
7
y
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0.500000
b = −0.500000
0 −2.25000
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u
3
+ u
2
− 1)
2
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
· (u
13
− 2u
12
+ ··· + 17u − 4)
c
2
(u + 1)(u
3
+ u
2
+ 2u + 1)
2
(u
5
+ 5u
4
+ 8u
3
+ 3u
2
− u + 1)
2
· (u
13
+ 8u
12
+ ··· + 209u + 16)
c
3
, c
7
u(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
(u
6
− 3u
4
+ 2u
2
+ 1)
· (u
13
+ 3u
12
+ ··· − 2u − 8)
c
4
(u + 1)(u
3
− u
2
+ 1)
2
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
· (u
13
− 2u
12
+ ··· + 17u − 4)
c
5
, c
6
(u + 1)(u
2
+ 1)
3
(u
10
+ 3u
9
+ ··· + 32u + 17)(u
13
− u
12
+ ··· + u − 1)
c
8
, c
9
, c
11
(u − 1)(u
2
+ 1)
3
(u
10
+ 3u
9
+ ··· + 32u + 17)(u
13
− u
12
+ ··· + u − 1)
c
10
((u − 1)
6
)(u + 1)(u
10
+ 11u
9
+ ··· + 1016u + 289)
· (u
13
+ 19u
12
+ ··· − 13u − 1)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y − 1)(y
3
− y
2
+ 2y − 1)
2
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
2
· (y
13
− 8y
12
+ ··· + 209y − 16)
c
2
(y − 1)(y
3
+ 3y
2
+ 2y − 1)
2
(y
5
− 9y
4
+ 32y
3
− 35y
2
− 5y − 1)
2
· (y
13
− 4y
12
+ ··· + 22817y − 256)
c
3
, c
7
y(y
3
− 3y
2
+ 2y + 1)
2
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
2
· (y
13
− 3y
12
+ ··· + 180y − 64)
c
5
, c
6
, c
8
c
9
, c
11
(y − 1)(y + 1)
6
(y
10
+ 11y
9
+ ··· + 1016y + 289)
· (y
13
+ 19y
12
+ ··· − 13y − 1)
c
10
((y − 1)
7
)(y
10
− 25y
9
+ ··· + 78660y + 83521)
· (y
13
− 49y
12
+ ··· − 61y − 1)
20
