11a
306
(K11a
306
)
A knot diagram
1
Linearized knot diagam
8 7 1 11 10 2 3 4 5 6 9
Solving Sequence
6,11
10 5 4 9 1 3 8 2 7
c
10
c
5
c
4
c
9
c
11
c
3
c
8
c
1
c
7
c
2
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hu
9
+ u
8
− 4u
7
− 4u
6
+ 4u
5
+ 4u
4
+ u
3
− u + 1i
I
u
2
= hu
42
+ u
41
+ ··· − 2u
4
+ 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
9
+ u
8
− 4u
7
− 4u
6
+ 4u
5
+ 4u
4
+ u
3
− u + 1i
(i) Arc colorings
a
6
=
0
u
a
11
=
1
0
a
10
=
1
u
2
a
5
=
−u
−u
3
+ u
a
4
=
u
3
− 2u
−u
3
+ u
a
9
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
u
6
− 3u
4
+ 2u
2
+ 1
u
8
− 4u
6
+ 4u
4
a
3
=
u
2
− 1
u
8
− 3u
6
− u
5
+ 2u
4
+ 2u
3
− u + 1
a
8
=
−u
3
+ 2u
u
8
− 3u
6
+ u
4
+ u
3
+ 2u
2
− 2u + 1
a
2
=
1
u
6
− 2u
4
− u
3
+ u − 1
a
7
=
u
u
7
− 2u
5
− u
4
+ u
2
a
7
=
u
u
7
− 2u
5
− u
4
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
7
− 4u
6
+ 16u
5
+ 12u
4
− 16u
3
− 8u
2
+ 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
9
− u
7
+ 2u
6
+ 8u
5
− 5u
4
− 5u
3
− 5u
2
+ 9u − 3
c
2
, c
5
, c
6
c
7
, c
9
, c
10
u
9
+ u
8
− 4u
7
− 4u
6
+ 4u
5
+ 4u
4
+ u
3
− u + 1
c
3
, c
11
u
9
− u
8
+ 4u
7
− 2u
6
+ 8u
5
− 6u
4
+ 9u
3
− 6u
2
+ 3u − 1
c
8
u
9
− 6u
8
+ 18u
7
− 33u
6
+ 39u
5
− 23u
4
− 7u
3
+ 26u
2
− 20u + 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
9
− 2y
8
+ 17y
7
− 30y
6
+ 112y
5
− 103y
4
+ 131y
3
− 145y
2
+ 51y −9
c
2
, c
5
, c
6
c
7
, c
9
, c
10
y
9
− 9y
8
+ 32y
7
− 54y
6
+ 38y
5
− 2y
4
+ y
3
− 10y
2
+ y −1
c
3
, c
11
y
9
+ 7y
8
+ 28y
7
+ 66y
6
+ 106y
5
+ 106y
4
+ 53y
3
+ 6y
2
− 3y −1
c
8
y
9
+ 6y
7
+ 25y
6
+ 23y
5
+ 17y
4
+ 213y
3
− 28y
2
− 16y −64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.287064 + 0.695105I
−1.39752 − 6.41727I 2.65899 + 8.21479I
u = −0.287064 − 0.695105I
−1.39752 + 6.41727I 2.65899 − 8.21479I
u = −1.30640
7.01397 12.1820
u = 0.423257 + 0.356395I
0.980950 + 0.551491I 9.15793 − 4.50455I
u = 0.423257 − 0.356395I
0.980950 − 0.551491I 9.15793 + 4.50455I
u = −1.42328 + 0.27641I
9.5593 − 13.5238I 11.6511 + 8.3193I
u = −1.42328 − 0.27641I
9.5593 + 13.5238I 11.6511 − 8.3193I
u = 1.44029 + 0.16872I
12.84680 + 4.88120I 15.4409 − 3.5107I
u = 1.44029 − 0.16872I
12.84680 − 4.88120I 15.4409 + 3.5107I
5
II. I
u
2
= hu
42
+ u
41
+ · · · − 2u
4
+ 1i
(i) Arc colorings
a
6
=
0
u
a
11
=
1
0
a
10
=
1
u
2
a
5
=
−u
−u
3
+ u
a
4
=
u
3
− 2u
−u
3
+ u
a
9
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
u
6
− 3u
4
+ 2u
2
+ 1
u
8
− 4u
6
+ 4u
4
a
3
=
−u
17
+ 8u
15
− 25u
13
+ 36u
11
− 19u
9
− 4u
7
+ 2u
5
+ 4u
3
− u
−u
19
+ 9u
17
− 32u
15
+ 55u
13
− 43u
11
+ 9u
9
+ 4u
5
− u
3
+ u
a
8
=
−u
10
+ 5u
8
− 8u
6
+ 3u
4
+ u
2
+ 1
u
10
− 4u
8
+ 5u
6
− 2u
4
+ u
2
a
2
=
u
28
− 13u
26
+ ··· + u
2
+ 1
−u
28
+ 12u
26
+ ··· + 2u
6
+ 3u
4
a
7
=
−u
41
− u
40
+ ··· + u + 2
u
41
− 19u
39
+ ··· + u − 1
a
7
=
−u
41
− u
40
+ ··· + u + 2
u
41
− 19u
39
+ ··· + u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
40
+ 72u
38
− 588u
36
+ 2864u
34
− 4u
33
− 9192u
32
+ 60u
31
+ 20240u
30
− 404u
29
−
30780u
28
+ 1600u
27
+ 31580u
26
− 4096u
25
− 20524u
24
+ 6988u
23
+ 7548u
22
− 7832u
21
−
1876u
20
+ 5336u
19
+ 1340u
18
−1704u
17
−664u
16
−16u
15
−192u
14
−116u
13
+ 212u
12
+
256u
11
− 28u
10
− 24u
9
− 44u
8
− 56u
7
+ 28u
6
+ 8u
5
+ 8u
4
+ 12u
3
+ 4u
2
− 8u + 6
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
42
− 3u
41
+ ··· − 2u − 1
c
2
, c
5
, c
6
c
7
, c
9
, c
10
u
42
+ u
41
+ ··· − 2u
4
+ 1
c
3
, c
11
u
42
− 9u
41
+ ··· − 920u + 113
c
8
(u
21
+ 3u
20
+ ··· + 4u − 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
42
− y
41
+ ··· − 24y + 1
c
2
, c
5
, c
6
c
7
, c
9
, c
10
y
42
− 37y
41
+ ··· − 4y
2
+ 1
c
3
, c
11
y
42
+ 11y
41
+ ··· + 84720y + 12769
c
8
(y
21
− 3y
20
+ ··· + 52y −1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.08927
1.57667 7.15490
u = −1.082920 + 0.161904I
−0.40568 − 3.16875I 2.95224 + 5.22442I
u = −1.082920 − 0.161904I
−0.40568 + 3.16875I 2.95224 − 5.22442I
u = 1.112720 + 0.206888I
4.64745 + 6.55351I 8.17560 − 6.03047I
u = 1.112720 − 0.206888I
4.64745 − 6.55351I 8.17560 + 6.03047I
u = 0.301718 + 0.707163I
4.04389 + 9.94224I 7.31059 − 8.24169I
u = 0.301718 − 0.707163I
4.04389 − 9.94224I 7.31059 + 8.24169I
u = 0.619519 + 0.389305I
5.30545 − 6.06326I 10.03226 + 2.92445I
u = 0.619519 − 0.389305I
5.30545 + 6.06326I 10.03226 − 2.92445I
u = −0.335269 + 0.641117I
6.08429 − 1.09840I 10.14786 + 3.17531I
u = −0.335269 − 0.641117I
6.08429 + 1.09840I 10.14786 − 3.17531I
u = 0.274697 + 0.655623I
−0.07785 + 2.71696I 5.48517 − 3.12164I
u = 0.274697 − 0.655623I
−0.07785 − 2.71696I 5.48517 + 3.12164I
u = 0.211792 + 0.670835I
−0.40568 + 3.16875I 2.95224 − 5.22442I
u = 0.211792 − 0.670835I
−0.40568 − 3.16875I 2.95224 + 5.22442I
u = −0.594417 + 0.333320I
−0.07785 + 2.71696I 5.48517 − 3.12164I
u = −0.594417 − 0.333320I
−0.07785 − 2.71696I 5.48517 + 3.12164I
u = 0.096884 + 0.668841I
1.62697 − 3.23317I 3.55215 + 1.92093I
u = 0.096884 − 0.668841I
1.62697 + 3.23317I 3.55215 − 1.92093I
u = −0.481440 + 0.468716I
6.75483 − 2.56601I 12.00469 + 3.90900I
u = −0.481440 − 0.468716I
6.75483 + 2.56601I 12.00469 − 3.90900I
u = −0.147288 + 0.653126I
−3.11833 −1.91795 + 0.I
u = −0.147288 − 0.653126I
−3.11833 −1.91795 + 0.I
u = −1.317380 + 0.229558I
6.02305 0
u = −1.317380 − 0.229558I
6.02305 0
u = 0.644973
1.57667 7.15490
u = 1.354040 + 0.243767I
1.62697 + 3.23317I 0
u = 1.354040 − 0.243767I
1.62697 − 3.23317I 0
u = −1.379260 + 0.261235I
4.64745 − 6.55351I 0
u = −1.379260 − 0.261235I
4.64745 + 6.55351I 0
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.41719 + 0.15750I
6.75483 − 2.56601I 0
u = −1.41719 − 0.15750I
6.75483 + 2.56601I 0
u = 1.42429 + 0.12838I
6.08429 − 1.09840I 0
u = 1.42429 − 0.12838I
6.08429 + 1.09840I 0
u = −1.40967 + 0.25849I
5.30545 − 6.06326I 0
u = −1.40967 − 0.25849I
5.30545 + 6.06326I 0
u = 1.41609 + 0.27243I
4.04389 + 9.94224I 0
u = 1.41609 − 0.27243I
4.04389 − 9.94224I 0
u = −1.44204 + 0.12357I
11.72580 + 4.35170I 0
u = −1.44204 − 0.12357I
11.72580 − 4.35170I 0
u = 1.42800 + 0.24722I
11.72580 + 4.35170I 0
u = 1.42800 − 0.24722I
11.72580 − 4.35170I 0
10
III. I
u
3
= hu − 1i
(i) Arc colorings
a
6
=
0
1
a
11
=
1
0
a
10
=
1
1
a
5
=
−1
0
a
4
=
−1
0
a
9
=
0
1
a
1
=
1
1
a
3
=
0
1
a
8
=
1
1
a
2
=
1
1
a
7
=
1
2
a
7
=
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
9
, c
10
, c
11
u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
c
2
, c
3
, c
5
c
6
, c
7
, c
8
c
9
, c
10
, c
11
y −1
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
4
u(u
9
− u
7
+ 2u
6
+ 8u
5
− 5u
4
− 5u
3
− 5u
2
+ 9u − 3)
· (u
42
− 3u
41
+ ··· − 2u − 1)
c
2
, c
5
, c
6
c
7
, c
9
, c
10
(u − 1)(u
9
+ u
8
− 4u
7
− 4u
6
+ 4u
5
+ 4u
4
+ u
3
− u + 1)
· (u
42
+ u
41
+ ··· − 2u
4
+ 1)
c
3
, c
11
(u − 1)(u
9
− u
8
+ 4u
7
− 2u
6
+ 8u
5
− 6u
4
+ 9u
3
− 6u
2
+ 3u − 1)
· (u
42
− 9u
41
+ ··· − 920u + 113)
c
8
(u − 1)(u
9
− 6u
8
+ ··· − 20u + 8)
· (u
21
+ 3u
20
+ ··· + 4u − 1)
2
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
· (y
9
− 2y
8
+ 17y
7
− 30y
6
+ 112y
5
− 103y
4
+ 131y
3
− 145y
2
+ 51y −9)
· (y
42
− y
41
+ ··· − 24y + 1)
c
2
, c
5
, c
6
c
7
, c
9
, c
10
(y −1)(y
9
− 9y
8
+ 32y
7
− 54y
6
+ 38y
5
− 2y
4
+ y
3
− 10y
2
+ y −1)
· (y
42
− 37y
41
+ ··· − 4y
2
+ 1)
c
3
, c
11
(y −1)(y
9
+ 7y
8
+ ··· − 3y −1)
· (y
42
+ 11y
41
+ ··· + 84720y + 12769)
c
8
(y −1)(y
9
+ 6y
7
+ ··· − 16y −64)
· (y
21
− 3y
20
+ ··· + 52y −1)
2
16
