11a
40
(K11a
40
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 10 11 3 7 5 6 9
Solving Sequence
5,10
6 11
2,7
4 1 9 8 3
c
5
c
10
c
6
c
4
c
1
c
9
c
8
c
3
c
2
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
45
− u
44
+ ··· + b + u, −u
45
− u
44
+ ··· + a + 1, u
46
+ 2u
45
+ ··· − u + 1i
I
u
2
= hb + 1, a + 2, u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 48 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
45
−u
44
+· · · + b+u, −u
45
−u
44
+· · · + a+1, u
46
+2u
45
+· · · − u+1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
11
=
−u
−u
3
+ u
a
2
=
u
45
+ u
44
+ ··· − 3u − 1
u
45
+ u
44
+ ··· − 2u
2
− u
a
7
=
−u
2
+ 1
−u
4
+ 2u
2
a
4
=
2u
45
+ 2u
44
+ ··· − 6u
3
− 3u
u
45
+ u
44
+ ··· − 2u
2
− 2u
a
1
=
u
5
− 2u
3
− u
u
5
− 3u
3
+ u
a
9
=
u
u
a
8
=
u
7
− 4u
5
+ 4u
3
u
9
− 5u
7
+ 7u
5
− 2u
3
+ u
a
3
=
−u
38
+ 21u
36
+ ··· − 2u − 1
u
45
+ u
44
+ ··· − 3u
2
− u
a
3
=
−u
38
+ 21u
36
+ ··· − 2u − 1
u
45
+ u
44
+ ··· − 3u
2
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
45
+ u
44
+ ··· + 8u − 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
46
− 3u
45
+ ··· − 4u + 1
c
2
u
46
+ 25u
45
+ ··· + 4u + 1
c
3
, c
7
u
46
+ u
45
+ ··· + 8u + 4
c
5
, c
6
, c
9
c
10
u
46
+ 2u
45
+ ··· − u + 1
c
8
u
46
− 15u
45
+ ··· − 232u + 16
c
11
u
46
− 14u
45
+ ··· − 885u + 207
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
46
− 25y
45
+ ··· − 4y + 1
c
2
y
46
− 5y
45
+ ··· − 44y + 1
c
3
, c
7
y
46
− 15y
45
+ ··· − 232y + 16
c
5
, c
6
, c
9
c
10
y
46
− 54y
45
+ ··· − 9y + 1
c
8
y
46
+ 29y
45
+ ··· − 9760y + 256
c
11
y
46
− 18y
45
+ ··· − 704565y + 42849
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.869521 + 0.344145I
a = 1.50416 + 0.73434I
b = 1.129120 + 0.456816I
−4.45409 − 3.41461I −10.48322 + 2.63232I
u = −0.869521 − 0.344145I
a = 1.50416 − 0.73434I
b = 1.129120 − 0.456816I
−4.45409 + 3.41461I −10.48322 − 2.63232I
u = 0.755545 + 0.498860I
a = 1.99178 − 1.61778I
b = 1.180690 + 0.538180I
−3.31037 − 10.45050I −8.69927 + 9.35917I
u = 0.755545 − 0.498860I
a = 1.99178 + 1.61778I
b = 1.180690 − 0.538180I
−3.31037 + 10.45050I −8.69927 − 9.35917I
u = 0.711638 + 0.469005I
a = −0.763674 − 0.318200I
b = 0.203438 − 0.815815I
−0.41848 − 5.45501I −5.35190 + 6.48052I
u = 0.711638 − 0.469005I
a = −0.763674 + 0.318200I
b = 0.203438 + 0.815815I
−0.41848 + 5.45501I −5.35190 − 6.48052I
u = −0.735390 + 0.424801I
a = −2.02595 − 2.04160I
b = −1.135150 + 0.447303I
−4.51433 + 4.40744I −10.64051 − 5.57891I
u = −0.735390 − 0.424801I
a = −2.02595 + 2.04160I
b = −1.135150 − 0.447303I
−4.51433 − 4.40744I −10.64051 + 5.57891I
u = 0.745322 + 0.383218I
a = −1.65094 + 0.77106I
b = −1.213930 + 0.324837I
−4.79298 − 1.73712I −11.07336 + 4.61384I
u = 0.745322 − 0.383218I
a = −1.65094 − 0.77106I
b = −1.213930 − 0.324837I
−4.79298 + 1.73712I −11.07336 − 4.61384I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.789174
a = 1.29622
b = 0.519026
−1.45882 −5.44130
u = −0.727976 + 0.286689I
a = 0.947922 − 0.230896I
b = −0.010899 − 0.557387I
−1.56660 + 0.49532I −7.68115 − 1.36018I
u = −0.727976 − 0.286689I
a = 0.947922 + 0.230896I
b = −0.010899 + 0.557387I
−1.56660 − 0.49532I −7.68115 + 1.36018I
u = 0.508676 + 0.507354I
a = 0.77267 − 1.48119I
b = 0.827592 + 0.600489I
2.79032 − 4.13635I −1.98154 + 7.56914I
u = 0.508676 − 0.507354I
a = 0.77267 + 1.48119I
b = 0.827592 − 0.600489I
2.79032 + 4.13635I −1.98154 − 7.56914I
u = 0.393201 + 0.514206I
a = −0.490341 + 0.227249I
b = 0.712026 − 0.604892I
3.12084 + 0.58749I −0.262566 + 0.327262I
u = 0.393201 − 0.514206I
a = −0.490341 − 0.227249I
b = 0.712026 + 0.604892I
3.12084 − 0.58749I −0.262566 − 0.327262I
u = 0.109343 + 0.629269I
a = 0.371133 + 0.758125I
b = 1.143360 − 0.519855I
−1.40155 + 6.64307I −5.04471 − 5.15805I
u = 0.109343 − 0.629269I
a = 0.371133 − 0.758125I
b = 1.143360 + 0.519855I
−1.40155 − 6.64307I −5.04471 + 5.15805I
u = 0.147821 + 0.548345I
a = 0.106291 − 0.656538I
b = 0.241472 + 0.712682I
1.22062 + 1.95597I −0.98131 − 1.36818I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.147821 − 0.548345I
a = 0.106291 + 0.656538I
b = 0.241472 − 0.712682I
1.22062 − 1.95597I −0.98131 + 1.36818I
u = −1.48057 + 0.06828I
a = 0.242649 + 0.138771I
b = 0.572969 + 0.701430I
−2.87293 + 1.27541I 0
u = −1.48057 − 0.06828I
a = 0.242649 − 0.138771I
b = 0.572969 − 0.701430I
−2.87293 − 1.27541I 0
u = −0.045501 + 0.511745I
a = −0.354500 + 1.284150I
b = −1.113980 − 0.361295I
−2.55505 − 1.20262I −6.54006 + 0.40776I
u = −0.045501 − 0.511745I
a = −0.354500 − 1.284150I
b = −1.113980 + 0.361295I
−2.55505 + 1.20262I −6.54006 − 0.40776I
u = −0.410393 + 0.283942I
a = 0.89569 − 1.72849I
b = −0.678138 + 0.225871I
−0.893062 + 1.082040I −6.48995 − 6.28251I
u = −0.410393 − 0.283942I
a = 0.89569 + 1.72849I
b = −0.678138 − 0.225871I
−0.893062 − 1.082040I −6.48995 + 6.28251I
u = −1.51614 + 0.11587I
a = 1.35148 + 0.70752I
b = 0.929112 − 0.623725I
−3.88727 + 6.30351I 0
u = −1.51614 − 0.11587I
a = 1.35148 − 0.70752I
b = 0.929112 + 0.623725I
−3.88727 − 6.30351I 0
u = 1.53678 + 0.03338I
a = −0.343145 + 1.028020I
b = −0.766719 − 0.461059I
−7.50480 − 1.94811I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.53678 − 0.03338I
a = −0.343145 − 1.028020I
b = −0.766719 + 0.461059I
−7.50480 + 1.94811I 0
u = −1.55032
a = −2.37917
b = −1.24159
−8.97468 0
u = 0.401298
a = −2.88943
b = −1.09496
−2.16597 1.80760
u = 1.61177 + 0.09154I
a = 0.560766 + 0.507385I
b = −0.121683 + 0.704360I
−9.57946 − 1.98178I 0
u = 1.61177 − 0.09154I
a = 0.560766 − 0.507385I
b = −0.121683 − 0.704360I
−9.57946 + 1.98178I 0
u = −1.61019 + 0.13498I
a = −0.394457 + 0.716202I
b = 0.178478 + 0.876402I
−8.32029 + 7.70926I 0
u = −1.61019 − 0.13498I
a = −0.394457 − 0.716202I
b = 0.178478 − 0.876402I
−8.32029 − 7.70926I 0
u = 1.61725 + 0.12164I
a = −2.39412 + 1.22565I
b = −1.163660 − 0.487582I
−12.55170 − 6.46011I 0
u = 1.61725 − 0.12164I
a = −2.39412 − 1.22565I
b = −1.163660 + 0.487582I
−12.55170 + 6.46011I 0
u = −1.61885 + 0.11009I
a = −2.28924 − 0.54856I
b = −1.267550 − 0.341014I
−12.88440 + 3.59966I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.61885 − 0.11009I
a = −2.28924 + 0.54856I
b = −1.267550 + 0.341014I
−12.88440 − 3.59966I 0
u = −1.62455 + 0.14574I
a = 2.45895 + 0.92240I
b = 1.209800 − 0.545440I
−11.4148 + 12.8867I 0
u = −1.62455 − 0.14574I
a = 2.45895 − 0.92240I
b = 1.209800 + 0.545440I
−11.4148 − 12.8867I 0
u = 1.64237
a = 1.72270
b = 0.784410
−9.96538 0
u = 1.64966 + 0.08726I
a = 2.12772 − 0.58397I
b = 1.160220 − 0.405425I
−13.13770 + 1.79965I 0
u = 1.64966 − 0.08726I
a = 2.12772 + 0.58397I
b = 1.160220 + 0.405425I
−13.13770 − 1.79965I 0
9
II. I
u
2
= hb + 1, a + 2, u
2
+ u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
−u + 1
a
11
=
−u
−u + 1
a
2
=
−2
−1
a
7
=
u
u
a
4
=
−1
−1
a
1
=
−1
0
a
9
=
u
u
a
8
=
u
u
a
3
=
−1
−1
a
3
=
−1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −17
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
2
c
2
, c
4
(u + 1)
2
c
3
, c
7
, c
8
u
2
c
5
, c
6
u
2
+ u − 1
c
9
, c
10
, c
11
u
2
− u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
2
c
3
, c
7
, c
8
y
2
c
5
, c
6
, c
9
c
10
, c
11
y
2
− 3y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −2.00000
b = −1.00000
−2.63189 −17.0000
u = −1.61803
a = −2.00000
b = −1.00000
−10.5276 −17.0000
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
2
)(u
46
− 3u
45
+ ··· − 4u + 1)
c
2
((u + 1)
2
)(u
46
+ 25u
45
+ ··· + 4u + 1)
c
3
, c
7
u
2
(u
46
+ u
45
+ ··· + 8u + 4)
c
4
((u + 1)
2
)(u
46
− 3u
45
+ ··· − 4u + 1)
c
5
, c
6
(u
2
+ u − 1)(u
46
+ 2u
45
+ ··· − u + 1)
c
8
u
2
(u
46
− 15u
45
+ ··· − 232u + 16)
c
9
, c
10
(u
2
− u − 1)(u
46
+ 2u
45
+ ··· − u + 1)
c
11
(u
2
− u − 1)(u
46
− 14u
45
+ ··· − 885u + 207)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
2
)(y
46
− 25y
45
+ ··· − 4y + 1)
c
2
((y −1)
2
)(y
46
− 5y
45
+ ··· − 44y + 1)
c
3
, c
7
y
2
(y
46
− 15y
45
+ ··· − 232y + 16)
c
5
, c
6
, c
9
c
10
(y
2
− 3y + 1)(y
46
− 54y
45
+ ··· − 9y + 1)
c
8
y
2
(y
46
+ 29y
45
+ ··· − 9760y + 256)
c
11
(y
2
− 3y + 1)(y
46
− 18y
45
+ ··· − 704565y + 42849)
15
