11a
175
(K11a
175
)
A knot diagram
1
Linearized knot diagam
6 1 8 11 10 2 3 4 5 9 7
Solving Sequence
2,6
7 1 3 8 4 9 11 10 5
c
6
c
1
c
2
c
7
c
3
c
8
c
11
c
10
c
5
c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
11
− u
10
− 2u
9
+ 2u
8
+ 3u
7
− 2u
6
− 2u
5
+ 2u
3
− u + 1i
I
u
2
= hu
40
− u
39
+ ··· − 3u
3
+ 1i
I
u
3
= hu + 1i
* 3 irreducible components of dim
C
= 0, with total 52 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
11
− u
10
− 2u
9
+ 2u
8
+ 3u
7
− 2u
6
− 2u
5
+ 2u
3
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
1
=
u
u
a
3
=
−u
3
−u
3
+ u
a
8
=
u
8
− u
6
+ u
4
+ 1
u
8
− 2u
6
+ 2u
4
a
4
=
u
10
− 3u
8
− u
7
+ 4u
6
+ 2u
5
− 2u
4
− 3u
3
+ u − 1
−u
8
+ 2u
6
− 2u
4
a
9
=
u
10
− u
8
− u
7
+ u
6
+ 2u
5
+ u
4
− u
3
u
3
− u
a
11
=
u
3
u
5
− u
3
+ u
a
10
=
−2u
10
+ u
9
+ 4u
8
− u
7
− 5u
6
− u
5
+ 2u
4
+ 3u
3
− u
2
+ 1
u
a
5
=
u
10
− 3u
8
− u
7
+ 5u
6
+ 2u
5
− 3u
4
− 3u
3
+ u − 1
−u
2
a
5
=
u
10
− 3u
8
− u
7
+ 5u
6
+ 2u
5
− 3u
4
− 3u
3
+ u − 1
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
10
+ 4u
9
−12u
8
−8u
7
+ 16u
6
+ 12u
5
−4u
4
−8u
3
−4u
2
+ 4u −2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
9
u
11
− u
10
− 2u
9
+ 2u
8
+ 3u
7
− 2u
6
− 2u
5
+ 2u
3
− u + 1
c
2
, c
10
u
11
+ 5u
10
+ ··· + u + 1
c
3
, c
7
, c
8
u
11
+ 4u
10
+ 3u
9
− 3u
8
+ 3u
7
+ 10u
6
− u
5
+ u
4
+ 6u
3
− 5u
2
+ 4
c
4
, c
11
u
11
+ u
9
+ 2u
8
+ 7u
7
+ u
6
+ 4u
5
− 3u
4
+ 12u
3
− 8u
2
+ 5u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
9
y
11
− 5y
10
+ ··· + y −1
c
2
, c
10
y
11
+ 3y
10
+ ··· − 7y −1
c
3
, c
7
, c
8
y
11
− 10y
10
+ ··· + 40y −16
c
4
, c
11
y
11
+ 2y
10
+ ··· + 41y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.472789 + 0.800775I
9.12060 − 3.24476I 5.98156 + 0.51441I
u = −0.472789 − 0.800775I
9.12060 + 3.24476I 5.98156 − 0.51441I
u = −0.912079
−1.65611 −5.73710
u = 1.054490 + 0.371149I
−4.87523 − 4.09967I −8.95070 + 5.15592I
u = 1.054490 − 0.371149I
−4.87523 + 4.09967I −8.95070 − 5.15592I
u = −1.081800 + 0.517146I
−2.76698 + 9.75515I −4.05162 − 10.29185I
u = −1.081800 − 0.517146I
−2.76698 − 9.75515I −4.05162 + 10.29185I
u = 1.094170 + 0.624458I
5.3908 − 13.9605I 0.53068 + 9.48051I
u = 1.094170 − 0.624458I
5.3908 + 13.9605I 0.53068 − 9.48051I
u = 0.361975 + 0.559972I
1.36102 + 0.98826I 4.35867 − 1.84291I
u = 0.361975 − 0.559972I
1.36102 − 0.98826I 4.35867 + 1.84291I
5
II. I
u
2
= hu
40
− u
39
+ · · · − 3u
3
+ 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
1
=
u
u
a
3
=
−u
3
−u
3
+ u
a
8
=
u
8
− u
6
+ u
4
+ 1
u
8
− 2u
6
+ 2u
4
a
4
=
u
13
− 2u
11
+ 3u
9
− 2u
7
+ 2u
5
− 2u
3
+ u
u
13
− 3u
11
+ 5u
9
− 4u
7
+ 2u
5
− u
3
+ u
a
9
=
−u
18
+ 3u
16
− 6u
14
+ 7u
12
− 7u
10
+ 7u
8
− 6u
6
+ 4u
4
− u
2
+ 1
−u
18
+ 4u
16
− 9u
14
+ 12u
12
− 11u
10
+ 8u
8
− 6u
6
+ 4u
4
− u
2
a
11
=
u
3
u
5
− u
3
+ u
a
10
=
u
39
− u
38
+ ··· + u
3
− 1
−u
38
+ 9u
36
+ ··· + u
3
− 1
a
5
=
−u
21
+ 4u
19
− 9u
17
+ 12u
15
− 10u
13
+ 6u
11
− 3u
9
+ 2u
7
+ u
5
− 2u
3
+ u
−u
23
+ 5u
21
+ ··· − 2u
3
+ u
a
5
=
−u
21
+ 4u
19
− 9u
17
+ 12u
15
− 10u
13
+ 6u
11
− 3u
9
+ 2u
7
+ u
5
− 2u
3
+ u
−u
23
+ 5u
21
+ ··· − 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
38
−32u
36
+ 140u
34
−416u
32
+ 928u
30
−1644u
28
+ 2412u
26
−3040u
24
+ 3380u
22
−
3364u
20
+ 2992u
18
+ 4u
17
− 2368u
16
− 16u
15
+ 1668u
14
+ 36u
13
− 1048u
12
− 52u
11
+
580u
10
+ 52u
9
− 268u
8
− 44u
7
+ 100u
6
+ 32u
5
− 28u
4
− 20u
3
+ 8u
2
+ 8u + 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
9
u
40
− u
39
+ ··· − 3u
3
+ 1
c
2
, c
10
u
40
+ 17u
39
+ ··· + 2u
2
+ 1
c
3
, c
7
, c
8
(u
20
− 2u
19
+ ··· − 2u + 1)
2
c
4
, c
11
u
40
− 3u
39
+ ··· + 6u + 3
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
9
y
40
− 17y
39
+ ··· + 2y
2
+ 1
c
2
, c
10
y
40
+ 11y
39
+ ··· + 4y + 1
c
3
, c
7
, c
8
(y
20
− 22y
19
+ ··· − 30y + 1)
2
c
4
, c
11
y
40
− 5y
39
+ ··· − 282y + 9
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.989179 + 0.332673I
−1.87696 + 1.08776I −3.66948 − 0.80831I
u = −0.989179 − 0.332673I
−1.87696 − 1.08776I −3.66948 + 0.80831I
u = −0.912778 + 0.528712I
0.112919 1.81750 + 0.I
u = −0.912778 − 0.528712I
0.112919 1.81750 + 0.I
u = 0.515254 + 0.788495I
7.58837 − 5.35722I 3.80298 + 4.77693I
u = 0.515254 − 0.788495I
7.58837 + 5.35722I 3.80298 − 4.77693I
u = −0.502219 + 0.792060I
9.28815 6.23474 + 0.I
u = −0.502219 − 0.792060I
9.28815 6.23474 + 0.I
u = 0.461488 + 0.804643I
7.28190 + 8.60190I 3.29856 − 5.07396I
u = 0.461488 − 0.804643I
7.28190 − 8.60190I 3.29856 + 5.07396I
u = 1.047750 + 0.294823I
−4.22715 + 2.78049I −7.53200 − 3.56896I
u = 1.047750 − 0.294823I
−4.22715 − 2.78049I −7.53200 + 3.56896I
u = 0.475874 + 0.769365I
3.57846 + 1.46542I 0.189647 − 0.302471I
u = 0.475874 − 0.769365I
3.57846 − 1.46542I 0.189647 + 0.302471I
u = 1.112840 + 0.027837I
3.57846 + 1.46542I 0.189647 − 0.302471I
u = 1.112840 − 0.027837I
3.57846 − 1.46542I 0.189647 + 0.302471I
u = 0.976421 + 0.536361I
0.79488 − 4.38017I 2.87668 + 6.69250I
u = 0.976421 − 0.536361I
0.79488 + 4.38017I 2.87668 − 6.69250I
u = −1.119580 + 0.049168I
1.78732 − 6.69475I −2.60998 + 4.97701I
u = −1.119580 − 0.049168I
1.78732 + 6.69475I −2.60998 − 4.97701I
u = −0.674204 + 0.548152I
0.79488 + 4.38017I 2.87668 − 6.69250I
u = −0.674204 − 0.548152I
0.79488 − 4.38017I 2.87668 + 6.69250I
u = −1.065390 + 0.469454I
−4.22715 + 2.78049I −7.53200 − 3.56896I
u = −1.065390 − 0.469454I
−4.22715 − 2.78049I −7.53200 + 3.56896I
u = 1.053770 + 0.517468I
−0.55874 − 5.32051I −0.06135 + 6.50240I
u = 1.053770 − 0.517468I
−0.55874 + 5.32051I −0.06135 − 6.50240I
u = 0.565990 + 0.536897I
1.96889 5.82360 + 0.I
u = 0.565990 − 0.536897I
1.96889 5.82360 + 0.I
u = 1.062890 + 0.635226I
5.95204 1.53406 + 0.I
u = 1.062890 − 0.635226I
5.95204 1.53406 + 0.I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.077540 + 0.613425I
1.78732 − 6.69475I −2.60998 + 4.97701I
u = 1.077540 − 0.613425I
1.78732 + 6.69475I −2.60998 − 4.97701I
u = −1.071010 + 0.632590I
7.58837 + 5.35722I 3.80298 − 4.77693I
u = −1.071010 − 0.632590I
7.58837 − 5.35722I 3.80298 + 4.77693I
u = −1.087960 + 0.626575I
7.28190 + 8.60190I 3.29856 − 5.07396I
u = −1.087960 − 0.626575I
7.28190 − 8.60190I 3.29856 + 5.07396I
u = −0.289073 + 0.622325I
−0.55874 − 5.32051I −0.06135 + 6.50240I
u = −0.289073 − 0.622325I
−0.55874 + 5.32051I −0.06135 − 6.50240I
u = −0.138437 + 0.513103I
−1.87696 + 1.08776I −3.66948 − 0.80831I
u = −0.138437 − 0.513103I
−1.87696 − 1.08776I −3.66948 + 0.80831I
10
III. I
u
3
= hu + 1i
(i) Arc colorings
a
2
=
0
−1
a
6
=
1
0
a
7
=
1
1
a
1
=
−1
−1
a
3
=
1
0
a
8
=
2
1
a
4
=
−1
−1
a
9
=
1
0
a
11
=
−1
−1
a
10
=
−2
−1
a
5
=
−1
−1
a
5
=
−1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
9
, c
10
u + 1
c
4
, c
11
u
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
9
, c
10
y −1
c
4
, c
11
y
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
−1.64493 −6.00000
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
9
(u + 1)(u
11
− u
10
− 2u
9
+ 2u
8
+ 3u
7
− 2u
6
− 2u
5
+ 2u
3
− u + 1)
· (u
40
− u
39
+ ··· − 3u
3
+ 1)
c
2
, c
10
(u + 1)(u
11
+ 5u
10
+ ··· + u + 1)(u
40
+ 17u
39
+ ··· + 2u
2
+ 1)
c
3
, c
7
, c
8
(u + 1)(u
11
+ 4u
10
+ ··· − 5u
2
+ 4)
· (u
20
− 2u
19
+ ··· − 2u + 1)
2
c
4
, c
11
u(u
11
+ u
9
+ 2u
8
+ 7u
7
+ u
6
+ 4u
5
− 3u
4
+ 12u
3
− 8u
2
+ 5u + 1)
· (u
40
− 3u
39
+ ··· + 6u + 3)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
9
(y −1)(y
11
− 5y
10
+ ··· + y −1)(y
40
− 17y
39
+ ··· + 2y
2
+ 1)
c
2
, c
10
(y −1)(y
11
+ 3y
10
+ ··· − 7y −1)(y
40
+ 11y
39
+ ··· + 4y + 1)
c
3
, c
7
, c
8
(y −1)(y
11
− 10y
10
+ ··· + 40y −16)(y
20
− 22y
19
+ ··· − 30y + 1)
2
c
4
, c
11
y(y
11
+ 2y
10
+ ··· + 41y −1)(y
40
− 5y
39
+ ··· − 282y + 9)
16
