11a
365
(K11a
365
)
A knot diagram
1
Linearized knot diagam
7 8 10 9 1 11 2 3 4 5 6
Solving Sequence
3,10
4 9 5 11 8 2 7 1 6
c
3
c
9
c
4
c
10
c
8
c
2
c
7
c
1
c
6
c
5
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
9
+ 4u
7
− u
6
+ 5u
5
− 3u
4
− 2u
2
− 3u + 1i
I
u
2
= hu
16
+ u
15
+ 6u
14
+ 6u
13
+ 15u
12
+ 15u
11
+ 17u
10
+ 17u
9
+ 4u
8
+ 4u
7
− 8u
6
− 8u
5
− 4u
4
− 4u
3
+ 2u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 25 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
= hu
9
+ 4u
7
− u
6
+ 5u
5
− 3u
4
− 2u
2
− 3u + 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
5
=
u
2
+ 1
u
4
+ 2u
2
a
11
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
8
=
u
3
+ 2u
u
3
+ u
a
2
=
−u
6
− 3u
4
− 2u
2
+ 1
−u
6
− 2u
4
− u
2
a
7
=
−u
6
− 3u
4
− 2u
2
+ 1
u
7
− u
6
+ 2u
5
− 3u
4
− 2u
2
− 2u + 1
a
1
=
u
3
+ 2u
u
8
− u
7
+ 3u
6
− 3u
5
+ 3u
4
− u
3
+ 3u − 1
a
6
=
−u
4
− u
2
+ 1
u
8
+ u
7
+ 2u
6
+ 2u
5
− u
4
− 3u
2
− 2u + 1
a
6
=
−u
4
− u
2
+ 1
u
8
+ u
7
+ 2u
6
+ 2u
5
− u
4
− 3u
2
− 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
8
+ 4u
7
+ 12u
6
+ 8u
5
+ 4u
4
− 16u
2
− 8u − 18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
8
, c
10
u
9
+ 3u
8
− u
7
− 8u
6
− u
5
+ 8u
4
+ 6u
3
− 7u − 2
c
3
, c
4
, c
5
c
6
, c
9
, c
11
u
9
+ 4u
7
+ u
6
+ 5u
5
+ 3u
4
+ 2u
2
− 3u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
c
8
, c
10
y
9
− 11y
8
+ 47y
7
− 98y
6
+ 103y
5
− 50y
4
+ 18y
3
− 52y
2
+ 49y −4
c
3
, c
4
, c
5
c
6
, c
9
, c
11
y
9
+ 8y
8
+ 26y
7
+ 39y
6
+ 13y
5
− 37y
4
− 40y
3
+ 2y
2
+ 13y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.930248
−14.7946 −18.2890
u = −0.092398 + 1.291150I
7.68628 + 2.63224I −3.26146 − 3.89078I
u = −0.092398 − 1.291150I
7.68628 − 2.63224I −3.26146 + 3.89078I
u = −0.704803
−4.75227 −19.3450
u = 0.285490 + 1.280780I
3.22608 − 7.14899I −8.72219 + 6.90579I
u = 0.285490 − 1.280780I
3.22608 + 7.14899I −8.72219 − 6.90579I
u = −0.445037 + 1.304010I
−6.66561 + 9.83268I −11.48734 − 5.80501I
u = −0.445037 − 1.304010I
−6.66561 − 9.83268I −11.48734 + 5.80501I
u = 0.278445
−0.461193 −21.4240
5
II. I
u
2
= hu
16
+ u
15
+ 6u
14
+ 6u
13
+ 15u
12
+ 15u
11
+ 17u
10
+ 17u
9
+ 4u
8
+
4u
7
− 8u
6
− 8u
5
− 4u
4
− 4u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
5
=
u
2
+ 1
u
4
+ 2u
2
a
11
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
8
=
u
3
+ 2u
u
3
+ u
a
2
=
−u
6
− 3u
4
− 2u
2
+ 1
−u
6
− 2u
4
− u
2
a
7
=
−u
9
− 4u
7
− 5u
5
+ 3u
−u
9
− 3u
7
− 3u
5
+ u
a
1
=
u
12
+ 5u
10
+ 9u
8
+ 4u
6
− 6u
4
− 5u
2
+ 1
u
12
+ 4u
10
+ 6u
8
+ 2u
6
− 3u
4
− 2u
2
a
6
=
2u
15
+ 12u
13
+ 29u
11
+ 28u
9
− 6u
7
− 30u
5
+ u
4
− 11u
3
+ 3u
2
+ 6u + 3
2u
15
+ 12u
13
+ ··· + 3u + 2
a
6
=
2u
15
+ 12u
13
+ 29u
11
+ 28u
9
− 6u
7
− 30u
5
+ u
4
− 11u
3
+ 3u
2
+ 6u + 3
2u
15
+ 12u
13
+ ··· + 3u + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
15
− 20u
13
− 40u
11
− 24u
9
+ 28u
7
+ 44u
5
+ 4u
3
− 12u − 14
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
8
, c
10
(u
8
− u
7
− 5u
6
+ 4u
5
+ 7u
4
− 4u
3
− 2u
2
+ 2u − 1)
2
c
3
, c
4
, c
5
c
6
, c
9
, c
11
u
16
− u
15
+ ··· − 2u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
c
8
, c
10
(y
8
− 11y
7
+ 47y
6
− 98y
5
+ 103y
4
− 50y
3
+ 6y
2
+ 1)
2
c
3
, c
4
, c
5
c
6
, c
9
, c
11
y
16
+ 11y
15
+ ··· + 12y
2
+ 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.926940 + 0.018527I
−10.78260 + 4.93524I −14.9844 − 2.9942I
u = −0.926940 − 0.018527I
−10.78260 − 4.93524I −14.9844 + 2.9942I
u = 0.289289 + 1.118510I
1.93558 −11.00319 + 0.I
u = 0.289289 − 1.118510I
1.93558 −11.00319 + 0.I
u = 0.076587 + 1.175000I
2.79859 − 1.27532I −9.18053 + 5.08518I
u = 0.076587 − 1.175000I
2.79859 + 1.27532I −9.18053 − 5.08518I
u = −0.300887 + 1.216990I
−1.05533 + 3.63283I −14.4224 − 4.5180I
u = −0.300887 − 1.216990I
−1.05533 − 3.63283I −14.4224 + 4.5180I
u = 0.695347 + 0.104492I
−1.05533 − 3.63283I −14.4224 + 4.5180I
u = 0.695347 − 0.104492I
−1.05533 + 3.63283I −14.4224 − 4.5180I
u = −0.457337 + 1.275720I
−6.88602 −11.82210 + 0.I
u = −0.457337 − 1.275720I
−6.88602 −11.82210 + 0.I
u = 0.453425 + 1.291550I
−10.78260 − 4.93524I −14.9844 + 2.9942I
u = 0.453425 − 1.291550I
−10.78260 + 4.93524I −14.9844 − 2.9942I
u = −0.329483 + 0.355718I
2.79859 + 1.27532I −9.18053 − 5.08518I
u = −0.329483 − 0.355718I
2.79859 − 1.27532I −9.18053 + 5.08518I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
8
, c
10
(u
8
− u
7
− 5u
6
+ 4u
5
+ 7u
4
− 4u
3
− 2u
2
+ 2u − 1)
2
· (u
9
+ 3u
8
− u
7
− 8u
6
− u
5
+ 8u
4
+ 6u
3
− 7u − 2)
c
3
, c
4
, c
5
c
6
, c
9
, c
11
(u
9
+ 4u
7
+ ··· − 3u − 1)(u
16
− u
15
+ ··· − 2u + 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
c
8
, c
10
(y
8
− 11y
7
+ 47y
6
− 98y
5
+ 103y
4
− 50y
3
+ 6y
2
+ 1)
2
· (y
9
− 11y
8
+ 47y
7
− 98y
6
+ 103y
5
− 50y
4
+ 18y
3
− 52y
2
+ 49y −4)
c
3
, c
4
, c
5
c
6
, c
9
, c
11
(y
9
+ 8y
8
+ 26y
7
+ 39y
6
+ 13y
5
− 37y
4
− 40y
3
+ 2y
2
+ 13y −1)
· (y
16
+ 11y
15
+ ··· + 12y
2
+ 1)
11
