11n
54
(K11n
54
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 9 10 3 1 11 7 6
Solving Sequence
6,10 3,7
4 11 1 2 5 9 8
c
6
c
3
c
10
c
11
c
2
c
4
c
9
c
8
c
1
, c
5
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
25
+ u
24
+ ··· + b − u, −u
25
+ u
24
+ ··· + a + 2, u
27
− 2u
26
+ ··· + 4u − 1i
I
u
2
= h−u
3
+ b + u + 1, −u
4
− u
3
+ u
2
+ a + u, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 33 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
25
+u
24
+· · ·+b−u, −u
25
+u
24
+· · ·+a+2, u
27
−2u
26
+· · ·+4u−1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
3
=
u
25
− u
24
+ ··· + 3u
2
− 2
u
25
− u
24
+ ··· + u
2
+ u
a
7
=
1
u
2
a
4
=
u
26
− u
25
+ ··· − 5u − 1
−u
26
+ u
25
+ ··· − 2u
2
+ 2u
a
11
=
−u
−u
3
+ u
a
1
=
−u
3
−u
3
+ u
a
2
=
−u
24
+ u
23
+ ··· − 3u − 1
−u
26
+ u
25
+ ··· − 5u
3
+ 2u
a
5
=
−u
8
+ u
6
− u
4
− 1
−u
10
+ 2u
8
− 3u
6
+ 2u
4
− u
2
a
9
=
u
3
u
5
− u
3
+ u
a
8
=
−u
11
+ 2u
9
− 2u
7
+ u
3
−u
11
+ 3u
9
− 4u
7
+ 3u
5
− u
3
+ u
a
8
=
−u
11
+ 2u
9
− 2u
7
+ u
3
−u
11
+ 3u
9
− 4u
7
+ 3u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
26
+ 6u
25
+ 20u
24
− 39u
23
− 47u
22
+ 126u
21
+ 51u
20
−
259u
19
+ 17u
18
+ 366u
17
− 150u
16
− 376u
15
+ 266u
14
+ 292u
13
− 287u
12
− 189u
11
+
231u
10
+ 112u
9
− 162u
8
− 61u
7
+ 97u
6
+ 36u
5
− 47u
4
− 23u
3
+ 17u
2
+ 12u − 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
27
− 7u
26
+ ··· − 5u + 1
c
2
u
27
+ 3u
26
+ ··· + 5u + 1
c
3
, c
7
u
27
+ u
26
+ ··· + 128u + 64
c
5
u
27
− 2u
26
+ ··· + 2u + 1
c
6
, c
10
u
27
+ 2u
26
+ ··· + 4u + 1
c
8
u
27
+ 8u
26
+ ··· + 16990u + 565
c
9
u
27
+ 12u
26
+ ··· + 12u + 1
c
11
u
27
+ 6u
26
+ ··· + 48u + 5
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
27
− 3y
26
+ ··· + 5y −1
c
2
y
27
+ 49y
26
+ ··· − 23y −1
c
3
, c
7
y
27
+ 39y
26
+ ··· − 28672y −4096
c
5
y
27
− 36y
26
+ ··· + 12y −1
c
6
, c
10
y
27
− 12y
26
+ ··· + 12y −1
c
8
y
27
− 72y
26
+ ··· + 171458760y −319225
c
9
y
27
+ 8y
26
+ ··· + 32y −1
c
11
y
27
− 4y
26
+ ··· + 504y −25
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.917142 + 0.407659I
a = 1.20846 + 2.04717I
b = 2.09938 + 1.19019I
−2.84916 + 1.60658I −4.88146 − 5.04321I
u = −0.917142 − 0.407659I
a = 1.20846 − 2.04717I
b = 2.09938 − 1.19019I
−2.84916 − 1.60658I −4.88146 + 5.04321I
u = 0.526875 + 0.831344I
a = 0.11316 − 2.41248I
b = 1.82470 − 0.93091I
12.17430 − 2.19817I −0.63423 + 2.08830I
u = 0.526875 − 0.831344I
a = 0.11316 + 2.41248I
b = 1.82470 + 0.93091I
12.17430 + 2.19817I −0.63423 − 2.08830I
u = 0.467388 + 0.843376I
a = −0.08288 + 2.25887I
b = −2.19783 + 0.76965I
11.82210 + 5.91141I −1.03607 − 2.33228I
u = 0.467388 − 0.843376I
a = −0.08288 − 2.25887I
b = −2.19783 − 0.76965I
11.82210 − 5.91141I −1.03607 + 2.33228I
u = 0.971756 + 0.498250I
a = 0.462169 − 0.060454I
b = −0.404355 + 0.874051I
−2.13464 − 3.70052I −7.14700 + 4.32876I
u = 0.971756 − 0.498250I
a = 0.462169 + 0.060454I
b = −0.404355 − 0.874051I
−2.13464 + 3.70052I −7.14700 − 4.32876I
u = 1.059940 + 0.286714I
a = −0.219785 − 0.584186I
b = 0.225820 − 0.093826I
−2.30226 − 0.55935I −5.41913 − 0.15707I
u = 1.059940 − 0.286714I
a = −0.219785 + 0.584186I
b = 0.225820 + 0.093826I
−2.30226 + 0.55935I −5.41913 + 0.15707I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.580852 + 0.656113I
a = −0.834314 − 0.778452I
b = −1.197700 − 0.124667I
2.76848 + 0.13713I 0.773547 − 0.780119I
u = −0.580852 − 0.656113I
a = −0.834314 + 0.778452I
b = −1.197700 + 0.124667I
2.76848 − 0.13713I 0.773547 + 0.780119I
u = −1.001280 + 0.594918I
a = −0.59412 − 1.46994I
b = −1.21967 − 0.98587I
1.53114 + 4.74698I −2.05877 − 5.37624I
u = −1.001280 − 0.594918I
a = −0.59412 + 1.46994I
b = −1.21967 + 0.98587I
1.53114 − 4.74698I −2.05877 + 5.37624I
u = −1.175710 + 0.039463I
a = −0.392154 − 0.491529I
b = −0.267337 + 1.188470I
5.98407 − 3.79755I −6.46791 + 2.18250I
u = −1.175710 − 0.039463I
a = −0.392154 + 0.491529I
b = −0.267337 − 1.188470I
5.98407 + 3.79755I −6.46791 − 2.18250I
u = −1.103060 + 0.538696I
a = 1.181940 − 0.254407I
b = 0.857869 − 0.770675I
−0.58075 + 6.65503I −3.43691 − 7.46005I
u = −1.103060 − 0.538696I
a = 1.181940 + 0.254407I
b = 0.857869 + 0.770675I
−0.58075 − 6.65503I −3.43691 + 7.46005I
u = −0.326760 + 0.690469I
a = −0.462660 + 0.572854I
b = 0.218748 + 0.864330I
1.66847 − 1.93992I 0.12713 + 2.72762I
u = −0.326760 − 0.690469I
a = −0.462660 − 0.572854I
b = 0.218748 − 0.864330I
1.66847 + 1.93992I 0.12713 − 2.72762I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.698871 + 0.307602I
a = −0.894710 + 0.463414I
b = 0.205039 − 0.148466I
−1.077410 − 0.093546I −6.21440 − 0.06252I
u = 0.698871 − 0.307602I
a = −0.894710 − 0.463414I
b = 0.205039 + 0.148466I
−1.077410 + 0.093546I −6.21440 + 0.06252I
u = 1.074550 + 0.661713I
a = 2.00442 − 1.14909I
b = 3.21245 − 0.01349I
10.52730 − 3.36992I −2.71394 + 2.50695I
u = 1.074550 − 0.661713I
a = 2.00442 + 1.14909I
b = 3.21245 + 0.01349I
10.52730 + 3.36992I −2.71394 − 2.50695I
u = 1.106510 + 0.642553I
a = −2.19489 + 1.59992I
b = −3.52700 + 0.08432I
9.8976 − 11.4401I −3.60368 + 6.65783I
u = 1.106510 − 0.642553I
a = −2.19489 − 1.59992I
b = −3.52700 − 0.08432I
9.8976 + 11.4401I −3.60368 − 6.65783I
u = 0.397840
a = −1.58927
b = 0.339752
−1.09734 −8.57440
7
II. I
u
2
= h−u
3
+ b + u + 1, −u
4
− u
3
+ u
2
+ a + u, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
3
=
u
4
+ u
3
− u
2
− u
u
3
− u − 1
a
7
=
1
u
2
a
4
=
u
4
+ u
3
− u
2
− u
u
3
− u − 1
a
11
=
−u
−u
3
+ u
a
1
=
−u
3
−u
3
+ u
a
2
=
u
4
− u
2
− u
−1
a
5
=
u
3
u
3
− u
a
9
=
u
3
u
5
− u
3
+ u
a
8
=
1
u
2
a
8
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
4
− 5u
2
− 5u − 5
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
6
c
2
, c
4
(u + 1)
6
c
3
, c
7
u
6
c
5
, c
8
, c
10
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
6
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
c
9
, c
11
u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
7
y
6
c
5
, c
6
, c
8
c
10
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
c
9
, c
11
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.002190 + 0.295542I
a = −0.685196 + 1.063260I
b = −1.258210 + 0.569162I
−3.53554 − 0.92430I −12.63596 − 0.09369I
u = 1.002190 − 0.295542I
a = −0.685196 − 1.063260I
b = −1.258210 − 0.569162I
−3.53554 + 0.92430I −12.63596 + 0.09369I
u = −0.428243 + 0.664531I
a = 0.917982 + 0.270708I
b = −0.082955 − 0.592379I
0.245672 − 0.924305I −2.59683 + 0.69886I
u = −0.428243 − 0.664531I
a = 0.917982 − 0.270708I
b = −0.082955 + 0.592379I
0.245672 + 0.924305I −2.59683 − 0.69886I
u = −1.073950 + 0.558752I
a = −0.732786 + 0.381252I
b = −0.158836 + 1.200140I
−1.64493 + 5.69302I −6.76721 − 4.86918I
u = −1.073950 − 0.558752I
a = −0.732786 − 0.381252I
b = −0.158836 − 1.200140I
−1.64493 − 5.69302I −6.76721 + 4.86918I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
27
− 7u
26
+ ··· − 5u + 1)
c
2
((u + 1)
6
)(u
27
+ 3u
26
+ ··· + 5u + 1)
c
3
, c
7
u
6
(u
27
+ u
26
+ ··· + 128u + 64)
c
4
((u + 1)
6
)(u
27
− 7u
26
+ ··· − 5u + 1)
c
5
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
27
− 2u
26
+ ··· + 2u + 1)
c
6
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)(u
27
+ 2u
26
+ ··· + 4u + 1)
c
8
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
27
+ 8u
26
+ ··· + 16990u + 565)
c
9
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)(u
27
+ 12u
26
+ ··· + 12u + 1)
c
10
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
27
+ 2u
26
+ ··· + 4u + 1)
c
11
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)(u
27
+ 6u
26
+ ··· + 48u + 5)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
6
)(y
27
− 3y
26
+ ··· + 5y −1)
c
2
((y −1)
6
)(y
27
+ 49y
26
+ ··· − 23y −1)
c
3
, c
7
y
6
(y
27
+ 39y
26
+ ··· − 28672y −4096)
c
5
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
27
− 36y
26
+ ··· + 12y −1)
c
6
, c
10
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
27
− 12y
26
+ ··· + 12y −1)
c
8
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
· (y
27
− 72y
26
+ ··· + 171458760y −319225)
c
9
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
27
+ 8y
26
+ ··· + 32y −1)
c
11
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
27
− 4y
26
+ ··· + 504y −25)
13
