11a
357
(K11a
357
)
A knot diagram
1
Linearized knot diagam
9 8 7 1 11 10 2 3 4 5 6
Solving Sequence
1,6
11 5 4 10 7 3 9 2 8
c
11
c
5
c
4
c
10
c
6
c
3
c
9
c
1
c
8
c
2
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
9
− 4u
7
+ 5u
5
+ u
2
− 3u − 1i
I
u
2
= hu
36
− u
35
+ ··· − 4u
3
+ 1i
* 2 irreducible components of dim
C
= 0, with total 45 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
+ 5u
5
+ u
2
− 3u − 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
11
=
1
−u
2
a
5
=
u
−u
3
+ u
a
4
=
−u
3
+ 2u
−u
3
+ u
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
7
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
3
=
u
8
− 3u
6
+ 3u
4
− u
3
− u
2
+ 2u
u
4
− 2u
2
a
9
=
−u
8
+ 3u
6
− 3u
4
− u
3
+ u + 1
−u
3
+ u
a
2
=
u
7
+ u
6
− 3u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u + 1
u
6
− 2u
4
+ u
2
a
8
=
−u
8
− u
7
+ 3u
6
+ 2u
5
− 2u
4
− u
2
− 2u
−u
5
+ u
3
+ u
a
8
=
−u
8
− u
7
+ 3u
6
+ 2u
5
− 2u
4
− u
2
− 2u
−u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
6
+ 4u
5
+ 12u
4
− 8u
3
− 8u
2
+ 4u − 18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
6
u
9
+ 4u
7
− 2u
6
+ 5u
5
− 6u
4
− 2u
3
− 5u
2
− 5u − 1
c
2
, c
5
, c
7
c
8
, c
10
, c
11
u
9
− 4u
7
+ 5u
5
+ u
2
− 3u − 1
c
9
u
9
+ 7u
8
+ 25u
7
+ 54u
6
+ 74u
5
+ 55u
4
+ u
3
− 42u
2
− 36u − 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
y
9
+ 8y
8
+ 26y
7
+ 32y
6
− 25y
5
− 116y
4
− 110y
3
− 17y
2
+ 15y −1
c
2
, c
5
, c
7
c
8
, c
10
, c
11
y
9
− 8y
8
+ 26y
7
− 40y
6
+ 19y
5
+ 24y
4
− 30y
3
− y
2
+ 11y −1
c
9
y
9
+ y
8
+ 17y
7
+ 16y
6
+ 102y
5
− 29y
4
+ 157y
3
− 956y
2
+ 624y −64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.098375 + 0.814801I
7.35406 − 4.61617I −4.22495 + 4.01969I
u = 0.098375 − 0.814801I
7.35406 + 4.61617I −4.22495 − 4.01969I
u = 1.18251
−5.71950 −15.9090
u = −1.188580 + 0.361061I
0.69960 + 3.87858I −10.64109 − 3.78555I
u = −1.188580 − 0.361061I
0.69960 − 3.87858I −10.64109 + 3.78555I
u = −1.37937
−11.3909 −21.8270
u = 1.341750 + 0.354713I
−1.71371 − 13.05000I −13.4391 + 8.3124I
u = 1.341750 − 0.354713I
−1.71371 + 13.05000I −13.4391 − 8.3124I
u = −0.306233
−0.504287 −19.6540
5
II. I
u
2
= hu
36
− u
35
+ · · · − 4u
3
+ 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
11
=
1
−u
2
a
5
=
u
−u
3
+ u
a
4
=
−u
3
+ 2u
−u
3
+ u
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
7
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
3
=
−u
15
+ 6u
13
− 14u
11
+ 14u
9
− 2u
7
− 6u
5
+ 2u
3
+ 2u
u
17
− 7u
15
+ 19u
13
− 22u
11
+ 3u
9
+ 14u
7
− 6u
5
− 4u
3
+ u
a
9
=
u
10
− 5u
8
+ 8u
6
− 3u
4
− 3u
2
+ 1
u
10
− 4u
8
+ 5u
6
− 3u
2
a
2
=
u
20
− 9u
18
+ ··· − 3u
2
+ 1
u
20
− 8u
18
+ 26u
16
− 40u
14
+ 19u
12
+ 24u
10
− 30u
8
+ 9u
4
a
8
=
2u
35
− u
34
+ ··· − 7u
2
+ 1
−u
34
+ 14u
32
+ ··· − 4u
2
+ 3u
a
8
=
2u
35
− u
34
+ ··· − 7u
2
+ 1
−u
34
+ 14u
32
+ ··· − 4u
2
+ 3u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
27
+ 44u
25
+ 4u
24
− 208u
23
− 40u
22
+ 532u
21
+ 172u
20
−
732u
19
− 400u
18
+ 348u
17
+ 504u
16
+ 416u
15
− 244u
14
− 628u
13
− 156u
12
+ 112u
11
+
224u
10
+ 208u
9
− 20u
8
− 40u
7
− 56u
6
− 48u
5
+ 4u
4
− 8u
3
+ 4u
2
− 4u − 10
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
6
u
36
+ 3u
35
+ ··· − 12u − 7
c
2
, c
5
, c
7
c
8
, c
10
, c
11
u
36
− u
35
+ ··· − 4u
3
+ 1
c
9
(u
18
− 3u
17
+ ··· − 7u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
y
36
+ 23y
35
+ ··· − 340y + 49
c
2
, c
5
, c
7
c
8
, c
10
, c
11
y
36
− 29y
35
+ ··· + 8y
2
+ 1
c
9
(y
18
+ 3y
17
+ ··· − 31y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.114880 + 0.814996I
2.86466 + 8.83442I −8.85054 − 6.32425I
u = −0.114880 − 0.814996I
2.86466 − 8.83442I −8.85054 + 6.32425I
u = −0.075687 + 0.812840I
4.10849 + 0.36044I −7.24415 − 0.04898I
u = −0.075687 − 0.812840I
4.10849 − 0.36044I −7.24415 + 0.04898I
u = −1.137600 + 0.360537I
−0.25332 − 4.56891I −11.76762 + 2.55639I
u = −1.137600 − 0.360537I
−0.25332 + 4.56891I −11.76762 − 2.55639I
u = 1.161330 + 0.360877I
4.10849 + 0.36044I −7.24415 − 0.04898I
u = 1.161330 − 0.360877I
4.10849 − 0.36044I −7.24415 + 0.04898I
u = −0.042366 + 0.732635I
2.49914 + 1.48028I −7.39740 − 4.69129I
u = −0.042366 − 0.732635I
2.49914 − 1.48028I −7.39740 + 4.69129I
u = 0.125186 + 0.707270I
−2.91493 − 2.96900I −12.88830 + 4.22200I
u = 0.125186 − 0.707270I
−2.91493 + 2.96900I −12.88830 − 4.22200I
u = −1.253710 + 0.284832I
−1.22218 + 2.17847I −11.24475 + 0.74332I
u = −1.253710 − 0.284832I
−1.22218 − 2.17847I −11.24475 − 0.74332I
u = 1.294410 + 0.195773I
−6.07645 −17.0816 + 0.I
u = 1.294410 − 0.195773I
−6.07645 −17.0816 + 0.I
u = 1.31676
−5.41700 −18.3110
u = 1.299400 + 0.312670I
−1.69882 − 5.26707I −12.9078 + 7.0444I
u = 1.299400 − 0.312670I
−1.69882 + 5.26707I −12.9078 − 7.0444I
u = −1.347930 + 0.085501I
−2.91493 + 2.96900I −12.88830 − 4.22200I
u = −1.347930 − 0.085501I
−2.91493 − 2.96900I −12.88830 + 4.22200I
u = 1.317490 + 0.356084I
−0.25332 − 4.56891I −11.76762 + 2.55639I
u = 1.317490 − 0.356084I
−0.25332 + 4.56891I −11.76762 − 2.55639I
u = −1.335910 + 0.303663I
−7.50591 + 6.65729I −18.0029 − 5.6815I
u = −1.335910 − 0.303663I
−7.50591 − 6.65729I −18.0029 + 5.6815I
u = 0.629383
−5.41700 −18.3110
u = −1.332130 + 0.356156I
2.86466 + 8.83442I −8.85054 − 6.32425I
u = −1.332130 − 0.356156I
2.86466 − 8.83442I −8.85054 + 6.32425I
u = 1.377060 + 0.080377I
−7.50591 − 6.65729I −18.0029 + 5.6815I
u = 1.377060 − 0.080377I
−7.50591 + 6.65729I −18.0029 − 5.6815I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.488414 + 0.379131I
−1.69882 + 5.26707I −12.9078 − 7.0444I
u = −0.488414 − 0.379131I
−1.69882 − 5.26707I −12.9078 + 7.0444I
u = 0.410957 + 0.392187I
2.49914 − 1.48028I −7.39740 + 4.69129I
u = 0.410957 − 0.392187I
2.49914 + 1.48028I −7.39740 − 4.69129I
u = −0.330280 + 0.456150I
−1.22218 − 2.17847I −11.24475 − 0.74332I
u = −0.330280 − 0.456150I
−1.22218 + 2.17847I −11.24475 + 0.74332I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
6
(u
9
+ 4u
7
− 2u
6
+ 5u
5
− 6u
4
− 2u
3
− 5u
2
− 5u − 1)
· (u
36
+ 3u
35
+ ··· − 12u − 7)
c
2
, c
5
, c
7
c
8
, c
10
, c
11
(u
9
− 4u
7
+ 5u
5
+ u
2
− 3u − 1)(u
36
− u
35
+ ··· − 4u
3
+ 1)
c
9
(u
9
+ 7u
8
+ 25u
7
+ 54u
6
+ 74u
5
+ 55u
4
+ u
3
− 42u
2
− 36u − 8)
· (u
18
− 3u
17
+ ··· − 7u + 1)
2
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
(y
9
+ 8y
8
+ 26y
7
+ 32y
6
− 25y
5
− 116y
4
− 110y
3
− 17y
2
+ 15y −1)
· (y
36
+ 23y
35
+ ··· − 340y + 49)
c
2
, c
5
, c
7
c
8
, c
10
, c
11
(y
9
− 8y
8
+ 26y
7
− 40y
6
+ 19y
5
+ 24y
4
− 30y
3
− y
2
+ 11y −1)
· (y
36
− 29y
35
+ ··· + 8y
2
+ 1)
c
9
(y
9
+ y
8
+ 17y
7
+ 16y
6
+ 102y
5
− 29y
4
+ 157y
3
− 956y
2
+ 624y −64)
· (y
18
+ 3y
17
+ ··· − 31y + 1)
2
12
