11a
357
(K11a
357
)
1
Arc Sequences
9 8 7 1 11 10 2 3 4 5 6
Solving Sequence
3,8
9 2 1 7 4 5 10 6 11
c
8
c
2
c
1
c
7
c
3
c
4
c
9
c
6
c
11
c
5
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
9
4u
7
+ 5u
5
+ u
2
3u 1i
I
u
2
= hu
36
u
35
+ ··· 4u
3
+ 1i
There are 2 irreducible components with 45 representations.
1
The knot diagram image is adapter from “C. Livingston and A. H. Moore, KnotInfo: Table of Knot
Invariants, http://www.indiana.edu/ knotinfo”
1
I. I
u
1
= hu
9
4u
7
+ 5u
5
+ u
2
3u 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
9
=
u
u
a
2
=
1
u
2
a
1
=
u
4
+ u
2
+ 1
u
4
2u
2
a
7
=
u
u
3
+ u
a
4
=
u
4
+ u
2
+ 1
u
6
2u
4
+ u
2
a
5
=
u
7
2u
5
u
4
+ u
2
+ u + 1
u
7
+ u
6
+ 3u
5
2u
4
2u
3
+ u
2
a
10
=
u
7
+ 2u
5
u
4
+ u
2
2u
u
7
+ u
6
3u
5
2u
4
+ 2u
3
+ u
2
+ u
a
6
=
u
8
+ u
7
3u
6
2u
5
+ 2u
4
+ u
2
+ 2u
u
7
+ 2u
5
+ u
4
+ u
3
u
2
3u 1
a
11
=
u
8
u
7
+ 2u
6
+ 3u
5
u
4
u
3
3u
u
8
2u
6
u
5
+ u
3
+ u
2
+ u
a
11
=
u
8
u
7
+ 2u
6
+ 3u
5
u
4
u
3
3u
u
8
2u
6
u
5
+ u
3
+ u
2
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.37937
11.3909 21.8268
u = 1.188582 0.361061I
0.69960 3.87858I 10.64109 + 3.78555I
u = 1.188582 + 0.361061I
0.69960 + 3.87858I 10.64109 3.78555I
u = 0.306233
0.504287 19.6540
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 = 1.341754 0.354713I
1.71371 + 13.05001I 13.4391 8.3124I
u = 1.341754 + 0.354713I
1.71371 13.05001I 13.4391 + 8.3124I
3
II. I
u
2
= hu
36
u
35
+ · · · 4u
3
+ 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
9
=
u
u
a
2
=
1
u
2
a
1
=
u
4
+ u
2
+ 1
u
4
2u
2
a
7
=
u
u
3
+ u
a
4
=
u
4
+ u
2
+ 1
u
6
2u
4
+ u
2
a
5
=
u
14
+ 5u
12
8u
10
+ u
8
+ 8u
6
4u
4
2u
2
+ 1
u
14
6u
12
+ 13u
10
10u
8
2u
6
+ 4u
4
+ u
2
a
10
=
u
11
4u
9
+ 4u
7
+ 2u
5
3u
3
2u
u
13
+ 5u
11
9u
9
+ 6u
7
u
3
+ u
a
6
=
u
21
8u
19
+ ··· + 6u
3
+ u
u
23
+ 9u
21
+ ··· 4u
3
+ u
a
11
=
u
35
14u
33
+ ··· 3u + 1
u
34
+ 14u
32
+ ··· + 2u 1
a
11
=
u
35
14u
33
+ ··· 3u + 1
u
34
+ 14u
32
+ ··· + 2u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.347925 0.085501I
2.91493 2.96900I 12.88830 + 4.22200I
u = 1.347925 + 0.085501I
2.91493 + 2.96900I 12.88830 4.22200I
u = 1.335909 0.303663I
7.50591 6.65729I 18.0029 + 5.6815I
u = 1.335909 + 0.303663I
7.50591 + 6.65729I 18.0029 5.6815I
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 = 1.253707 0.284832I
1.22218 2.17847I 11.24475 0.74332I
u = 1.253707 + 0.284832I
1.22218 + 2.17847I 11.24475 + 0.74332I
u = 1.137595 0.360537I
0.25332 + 4.56891I 11.76762 2.55639I
u = 1.137595 + 0.360537I
0.25332 4.56891I 11.76762 + 2.55639I
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 = 0.330280 0.456150I
1.22218 + 2.17847I 11.24475 + 0.74332I
u = 0.330280 + 0.456150I
1.22218 2.17847I 11.24475 0.74332I
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 = 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 = 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 = 0.629383
5.41700 18.3114
u = 1.161329 0.360877I
4.10849 0.36044I 7.24415 + 0.04898I
u = 1.161329 + 0.360877I
4.10849 + 0.36044I 7.24415 0.04898I
u = 1.294406 0.195773I
6.07645 17.0816
u = 1.294406 + 0.195773I
6.07645 17.0816
u = 1.299395 0.312670I
1.69882 + 5.26707I 12.9078 7.0444I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.299395 + 0.312670I
1.69882 5.26707I 12.9078 + 7.0444I
u = 1.31676
5.41700 18.3114
u = 1.317493 0.356084I
0.25332 + 4.56891I 11.76762 2.55639I
u = 1.317493 + 0.356084I
0.25332 4.56891I 11.76762 + 2.55639I
u = 1.377056 0.080377I
7.50591 + 6.65729I 18.0029 5.6815I
u = 1.377056 + 0.080377I
7.50591 6.65729I 18.0029 + 5.6815I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
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
+ ··· 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)
(1 7u + 9u
2
13u
3
+ 9u
4
u
5
+ 4u
6
9u
7
+ 22u
8
25u
9
+ 21u
10
31u
11
+ 44u
12
30u
13
+ 16u
14
7u
15
+ 6u
16
3u
17
+ u
18
)
2
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
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)
(1 31y 83y
2
13y
3
+ 45y
4
75y
5
224y
6
+ 21y
7
+ 354y
8
+ 207y
9
+ 449y
10
+ 25y
11
+ 616y
12
+ 220y
13
+ 220y
14
+ 51y
15
+ 26y
16
+ 3y
17
+ y
18
)
2
7